From A First Course in Linear Algebra
Version 2.11
© 2004.
Licensed under the GNU Free Documentation License.
http://linear.ups.edu/
A vector space is defined as a set with two operations, meeting ten properties
(Definition VS). Just as the definition of span of a set of vectors only required
knowing how to add vectors and how to multiply vectors by scalars, so it is with
linear independence. A definition of a linear independent set of vectors in an
arbitrary vector space only requires knowing how to form linear combinations
and equating these with the zero vector. Since every vector space must
have a zero vector (Property Z), we always have a zero vector at our
disposal.
In this section we will also put a twist on the notion of the span of a set of vectors. Rather than beginning with a set of vectors and creating a subspace that is the span, we will instead begin with a subspace and look for a set of vectors whose span equals the subspace.
The combination of linear independence and spanning will be very important going forward.
Our previous definition of linear independence (Definition LI) employed a relation of linear dependence that was a linear combination on one side of an equality and a zero vector on the other side. As a linear combination in a vector space (Definition LC) depends only on vector addition and scalar multiplication, and every vector space must have a zero vector (Property Z), we can extend our definition of linear independence from the setting of {ℂ}^{m} to the setting of a general vector space V with almost no changes. Compare these next two definitions with Definition RLDCV and Definition LICV.
Definition RLD
Relation of Linear Dependence
Suppose that V is a vector
space. Given a set of vectors S = \left \{{u}_{1},\kern 1.95872pt {u}_{2},\kern 1.95872pt {u}_{3},\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {u}_{n}\right \},
an equation of the form
{α}_{1}{u}_{1} + {α}_{2}{u}_{2} + {α}_{3}{u}_{3} + \mathrel{⋯} + {α}_{n}{u}_{n} = 0

is a relation of linear dependence on S. If this equation is formed in a trivial fashion, i.e. {α}_{i} = 0, 1 ≤ i ≤ n, then we say it is a trivial relation of linear dependence on S. △
Definition LI
Linear Independence
Suppose that V is a vector
space. The set of vectors S = \left \{{u}_{1},\kern 1.95872pt {u}_{2},\kern 1.95872pt {u}_{3},\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {u}_{n}\right \}
from V
is linearly dependent if there is a relation of linear dependence on
S that
is not trivial. In the case where the only relation of linear dependence on
S is the trivial one, then
S is a linearly independent
set of vectors. △
Notice the emphasis on the word “only.” This might remind you of the definition of a nonsingular matrix, where if the matrix is employed as the coefficient matrix of a homogeneous system then the only solution is the trivial one.
Example LIP4
Linear independence in {P}_{4}
In the vector space of polynomials with degree 4 or less,
{P}_{4}
(Example VSP) consider the set
S = \left \{2{x}^{4} + 3{x}^{3} + 2{x}^{2} − x + 10,\kern 1.95872pt − {x}^{4} − 2{x}^{3} + {x}^{2} + 5x − 8,\kern 1.95872pt 2{x}^{4} + {x}^{3} + 10{x}^{2} + 17x − 2\right \}.

Is this set of vectors linearly independent or dependent? Consider that
This is a nontrivial relation of linear dependence (Definition RLD) on the set S and so convinces us that S is linearly dependent (Definition LI).
Now, I hear you say, “Where did those scalars come from?” Do not worry about that right now, just be sure you understand why the above explanation is sufficient to prove that S is linearly dependent. The remainder of the example will demonstrate how we might find these scalars if they had not been provided so readily. Let’s look at another set of vectors (polynomials) from {P}_{4}. Let
Suppose we have a relation of linear dependence on this set,
Using our definitions of vector addition and scalar multiplication in {P}_{4} (Example VSP), we arrive at,
Equating coefficients, we arrive at the homogeneous system of equations,
We form the coefficient matrix of this homogeneous system of equations and rowreduce to find
\left [\array{
\text{1}&0&0&0\cr
0&\text{1 } &0 &0
\cr
0&0&\text{1}&0\cr
0&0 &0 &\text{1}
\cr
0&0&0&0 } \right ]

We expected the system to be consistent (Theorem HSC) and so can compute n − r = 4 − 4 = 0 and Theorem CSRN tells us that the solution is unique. Since this is a homogeneous system, this unique solution is the trivial solution (Definition TSHSE), {α}_{1} = 0, {α}_{2} = 0, {α}_{3} = 0, {α}_{4} = 0. So by Definition LI the set T is linearly independent.
A few observations. If we had discovered infinitely many solutions, then we could have used one of the nontrivial ones to provide a linear combination in the manner we used to show that S was linearly dependent. It is important to realize that it is not interesting that we can create a relation of linear dependence with zero scalars — we can always do that — but that for T, this is the only way to create a relation of linear dependence. It was no accident that we arrived at a homogeneous system of equations in this example, it is related to our use of the zero vector in defining a relation of linear dependence. It is easy to present a convincing statement that a set is linearly dependent (just exhibit a nontrivial relation of linear dependence) but a convincing statement of linear independence requires demonstrating that there is no relation of linear dependence other than the trivial one. Notice how we relied on theorems from Chapter SLE to provide this demonstration. Whew! There’s a lot going on in this example. Spend some time with it, we’ll be waiting patiently right here when you get back. ⊠
Example LIM32
Linear independence in {M}_{32}
Consider the two sets of vectors R
and S from the
vector space of all 3 × 2
matrices, {M}_{32}
(Example VSM)
One set is linearly independent, the other is not. Which is which? Let’s examine R first. Build a generic relation of linear dependence (Definition RLD),
{
α}_{1}\left [\array{
3&−1\cr
1& 4
\cr
6&−6 } \right ]+{α}_{2}\left [\array{
−2& 3\cr
1 &−3
\cr
−2&−6 } \right ]+{α}_{3}\left [\array{
6 &−6\cr
−1 & 0
\cr
7 &−9 } \right ]+{α}_{4}\left [\array{
7 & 9\cr
−4 &−5
\cr
2 & 5 } \right ] = 0

Massaging the lefthand side with our definitions of vector addition and scalar multiplication in {M}_{32} (Example VSM) we obtain,
\left [\array{
3{α}_{1} − 2{α}_{2} + 6{α}_{3} + 7{α}_{4}&−1{α}_{1} + 3{α}_{2} − 6{α}_{3} + 9{α}_{4}
\cr
1{α}_{1} + 1{α}_{2} − {α}_{3} − 4{α}_{4} & 4{α}_{1} − 3{α}_{2} + −5{α}_{4}
\cr
6{α}_{1} − 2{α}_{2} + 7{α}_{3} + 2{α}_{4}&−6{α}_{1} − 6{α}_{2} − 9{α}_{3} + 5{α}_{4} } \right ] = \left [\array{
0&0\cr
0&0
\cr
0&0 } \right ]

Using our definition of matrix equality (Definition ME) and equating corresponding entries we get the homogeneous system of six equations in four variables,
Form the coefficient matrix of this homogeneous system and rowreduce to obtain
\left [\array{
\text{1}&0&0&0\cr
0&\text{1 } &0 &0
\cr
0&0&\text{1}&0\cr
0&0 &0 &\text{1}
\cr
0&0&0&0\cr
0&0 &0 &0 } \right ]

Analyzing this matrix we are led to conclude that {α}_{1} = 0, {α}_{2} = 0, {α}_{3} = 0, {α}_{4} = 0. This means there is only a trivial relation of linear dependence on the vectors of R and so we call R a linearly independent set (Definition LI).
So it must be that S is linearly dependent. Let’s see if we can find a nontrivial relation of linear dependence on S. We will begin as with R, by constructing a relation of linear dependence (Definition RLD) with unknown scalars,
{
α}_{1}\left [\array{
2& 0\cr
1&−1
\cr
1& 3} \right ]+{α}_{2}\left [\array{
−4& 0\cr
−2 & 2
\cr
−2&−6 } \right ]+{α}_{3}\left [\array{
1 &1\cr
−2 &1
\cr
2 &4 } \right ]+{α}_{4}\left [\array{
−5 &3\cr
−10 &7
\cr
2 &0 } \right ] = 0

Massaging the lefthand side with our definitions of vector addition and scalar multiplication in {M}_{32} (Example VSM) we obtain,
\left [\array{
2{α}_{1} − 4{α}_{2} + {α}_{3} − 5{α}_{4} & {α}_{3} + 3{α}_{4}
\cr
{α}_{1} − 2{α}_{2} − 2{α}_{3} − 10{α}_{4}&−{α}_{1} + 2{α}_{2} + {α}_{3} + 7{α}_{4}
\cr
{α}_{1} − 2{α}_{2} + 2{α}_{3} + 2{α}_{4} & 3{α}_{1} − 6{α}_{2} + 4{α}_{3} } \right ] = \left [\array{
0&0\cr
0&0
\cr
0&0 } \right ]

Using our definition of matrix equality (Definition ME) and equating corresponding entries we get the homogeneous system of six equations in four variables,
Form the coefficient matrix of this homogeneous system and rowreduce to obtain
\left [\array{
\text{1}&−2&0&−4\cr
0& 0 &\text{1 } & 3
\cr
0& 0 &0& 0\cr
0& 0 &0 & 0
\cr
0& 0 &0& 0\cr
0& 0 &0 & 0 } \right ]

Analyzing this we see that the system is consistent (we expected this since the system is homogeneous, Theorem HSC) and has n − r = 4 − 2 = 2 free variables, namely {α}_{2} and {α}_{4}. This means there are infinitely many solutions, and in particular, we can find a nontrivial solution, so long as we do not pick all of our free variables to be zero. The mere presence of a nontrivial solution for these scalars is enough to conclude that S is a linearly dependent set (Definition LI). But let’s go ahead and explicitly construct a nontrivial relation of linear dependence.
Choose {α}_{2} = 1 and {α}_{4} = −1. There is nothing special about this choice, there are infinitely many possibilities, some “easier” than this one, just avoid picking both variables to be zero. Then we find the corresponding dependent variables to be {α}_{1} = −2 and {α}_{3} = 3. So the relation of linear dependence,
(−2)\left [\array{
2& 0\cr
1&−1
\cr
1& 3} \right ]+(1)\left [\array{
−4& 0\cr
−2 & 2
\cr
−2&−6 } \right ]+(3)\left [\array{
1 &1\cr
−2 &1
\cr
2 &4 } \right ]+(−1)\left [\array{
−5 &3\cr
−10 &7
\cr
2 &0 } \right ] = \left [\array{
0&0\cr
0&0
\cr
0&0 } \right ]

is an ironclad demonstration that S is linearly dependent. Can you construct another such demonstration? ⊠
Example LIC
Linearly independent set in the crazy vector space
Is the set R = \left \{(1,\kern 1.95872pt 0),\kern 1.95872pt (6,\kern 1.95872pt 3)\right \}
linearly independent in the crazy vector space
C
(Example CVS)? We begin with an arbitrary relation of linear independence on
R
and then massage it to a point where we can apply the definition of equality in C. Recall the definitions of vector addition and scalar multiplication in C are not what you would expect.
Equality in C (Example CVS) then yields the two equations,
which becomes the homogeneous system
Since the coefficient matrix of this system is nonsingular (check this!) the system has only the trivial solution {a}_{1} = {a}_{2} = 0. By Definition LI the set R is linearly independent. Notice that even though the zero vector of C is not what we might first suspected, a question about linear independence still concludes with a question about a homogeneous system of equations. Hmmm. ⊠
In a vector space V , suppose we are given a set of vectors S ⊆ V . Then we can immediately construct a subspace, \left \langle S\right \rangle , using Definition SS and then be assured by Theorem SSS that the construction does provide a subspace. We now turn the situation upsidedown. Suppose we are first given a subspace W ⊆ V . Can we find a set S so that \left \langle S\right \rangle = W? Typically W is infinite and we are searching for a finite set of vectors S that we can combine in linear combinations and “build” all of W.
I like to think of S as the raw materials that are sufficient for the construction of W. If you have nails, lumber, wire, copper pipe, drywall, plywood, carpet, shingles, paint (and a few other things), then you can combine them in many different ways to create a house (or infinitely many different houses for that matter). A fastfood restaurant may have beef, chicken, beans, cheese, tortillas, taco shells and hot sauce and from this small list of ingredients build a wide variety of items for sale. Or maybe a better analogy comes from Ben Cordes — the additive primary colors (red, green and blue) can be combined to create many different colors by varying the intensity of each. The intensity is like a scalar multiple, and the combination of the three intensities is like vector addition. The three individual colors, red, green and blue, are the elements of the spanning set.
Because we will use terms like “spanned by” and “spanning set,” there is the potential for confusion with “the span.” Come back and reread the first paragraph of this subsection whenever you are uncertain about the difference. Here’s the working definition.
Definition TSVS
To Span a Vector Space
Suppose V is a vector
space. A subset S
of V is a spanning
set for V if
\left \langle S\right \rangle = V . In this case,
we also say S
spans V .
△
The definition of a spanning set requires that two sets (subspaces actually) be equal. If S is a subset of V , then \left \langle S\right \rangle ⊆ V , always. Thus it is usually only necessary to prove that V ⊆\left \langle S\right \rangle . Now would be a good time to review Definition SE.
Example SSP4
Spanning set in {P}_{4}
In Example SP4 we showed that
W = \left \{p(x)\mathrel{∣}p ∈ {P}_{4},\ p(2) = 0\right \}

is a subspace of {P}_{4}, the vector space of polynomials with degree at most 4 (Example VSP). In this example, we will show that the set
S = \left \{x − 2,\kern 1.95872pt {x}^{2} − 4x + 4,\kern 1.95872pt {x}^{3} − 6{x}^{2} + 12x − 8,\kern 1.95872pt {x}^{4} − 8{x}^{3} + 24{x}^{2} − 32x + 16\right \}

is a spanning set for W. To do this, we require that W = \left \langle S\right \rangle . This is an equality of sets. We can check that every polynomial in S has x = 2 as a root and therefore S ⊆ W. Since W is closed under addition and scalar multiplication, \left \langle S\right \rangle ⊆ W also.
So it remains to show that W ⊆\left \langle S\right \rangle (Definition SE). To do this, begin by choosing an arbitrary polynomial in W, say r(x) = a{x}^{4} + b{x}^{3} + c{x}^{2} + dx + e ∈ W. This polynomial is not as arbitrary as it would appear, since we also know it must have x = 2 as a root. This translates to
0 = a{(2)}^{4} + b{(2)}^{3} + c{(2)}^{2} + d(2) + e = 16a + 8b + 4c + 2d + e

as a condition on r.
We wish to show that r is a polynomial in \left \langle S\right \rangle , that is, we want to show that r can be written as a linear combination of the vectors (polynomials) in S. So let’s try.
Equating coefficients (vector equality in {P}_{4}) gives the system of five equations in four variables,
Any solution to this system of equations will provide the linear combination we need to determine if r ∈\left \langle S\right \rangle , but we need to be convinced there is a solution for any values of a,\kern 1.95872pt b,\kern 1.95872pt c,\kern 1.95872pt d,\kern 1.95872pt e that qualify r to be a member of W. So the question is: is this system of equations consistent? We will form the augmented matrix, and rowreduce. (We probably need to do this by hand, since the matrix is symbolic — reversing the order of the first four rows is the best way to start). We obtain a matrix in reduced rowechelon form
\left [\array{
\text{1}&0&0&0& 32a + 12b + 4c + d
\cr
0&\text{1}&0&0& 24a + 6b + c
\cr
0&0&\text{1}&0& 8a + b\cr
0&0 &0 &\text{1 } & a
\cr
0&0&0&0&16a + 8b + 4c + 2d + e } \right ] = \left [\array{
\text{1}&0&0&0&32a + 12b + 4c + d
\cr
0&\text{1}&0&0& 24a + 6b + c
\cr
0&0&\text{1}&0& 8a + b\cr
0&0 &0 &\text{1 } & a
\cr
0&0&0&0& 0 } \right ]

For your results to match our first matrix, you may find it necessary to multiply the final row of your rowreduced matrix by the appropriate scalar, and/or add multiples of this row to some of the other rows. To obtain the second version of the matrix, the last entry of the last column has been simplified to zero according to the one condition we were able to impose on an arbitrary polynomial from W. So with no leading 1’s in the last column, Theorem RCLS tells us this system is consistent. Therefore, any polynomial from W can be written as a linear combination of the polynomials in S, so W ⊆\left \langle S\right \rangle . Therefore, W = \left \langle S\right \rangle and S is a spanning set for W by Definition TSVS.
Notice that an alternative to rowreducing the augmented matrix by hand would be to appeal to Theorem FS by expressing the column space of the coefficient matrix as a null space, and then verifying that the condition on r guarantees that r is in the column space, thus implying that the system is always consistent. Give it a try, we’ll wait. This has been a complicated example, but worth studying carefully. ⊠
Given a subspace and a set of vectors, as in Example SSP4 it can take some work to determine that the set actually is a spanning set. An even harder problem is to be confronted with a subspace and required to construct a spanning set with no guidance. We will now work an example of this flavor, but some of the steps will be unmotivated. Fortunately, we will have some better tools for this type of problem later on.
Example SSM22
Spanning set in {M}_{22}
In the space of all 2 × 2
matrices, {M}_{22}
consider the subspace
Z = \left \{\left [\array{
a&b\cr
c&d } \right ]\mathrel{∣}a + 3b − c − 5d = 0,\ −2a − 6b + 3c + 14d = 0\right \}

and find a spanning set for Z.
We need to construct a limited number of matrices in Z so that every matrix in Z can be expressed as a linear combination of this limited number of matrices. Suppose that B = \left [\array{ a&b\cr c&d } \right ] is a matrix in Z. Then we can form a column vector with the entries of B and write
\left [\array{
a\cr
b
\cr
c\cr
d } \right ] ∈N\kern 1.95872pt \left (\left [\array{
1 & 3 &−1&−5\cr
−2 &−6 & 3 & 14 } \right ]\right )

Rowreducing this matrix and applying Theorem REMES we obtain the equivalent statement,
\left [\array{
a\cr
b
\cr
c\cr
d } \right ] ∈N\kern 1.95872pt \left (\left [\array{
\text{1}&3&0&−1\cr
0&0 &\text{1 } & 4 } \right ]\right )

We can then express the subspace Z in the following equal forms,
So the set
Q = \left \{\left [\array{
−3&1\cr
0 &0 } \right ],\kern 1.95872pt \left [\array{
1 &0\cr
−4 &1 } \right ]\right \}

spans Z by Definition TSVS. ⊠
Example SSC
Spanning set in the crazy vector space
In Example LIC we determined that the set
R = \left \{(1,\kern 1.95872pt 0),\kern 1.95872pt (6,\kern 1.95872pt 3)\right \}
is linearly independent in the crazy vector space
C (Example CVS).
We now show that R is
a spanning set for C.
Given an arbitrary vector (x,\kern 1.95872pt y) ∈ C we desire to show that it can be written as a linear combination of the elements of R. In other words, are there scalars {a}_{1} and {a}_{2} so that
(x,\kern 1.95872pt y) = {a}_{1}(1,\kern 1.95872pt 0) + {a}_{2}(6,\kern 1.95872pt 3)

We will act as if this equation is true and try to determine just what {a}_{1} and {a}_{2} would be (as functions of x and y).
Equality in C then yields the two equations,
which becomes the linear system with a matrix representation
\left [\array{
2&7\cr
1&4 } \right ]\left [\array{
{a}_{1}
\cr
{a}_{2} } \right ] = \left [\array{
x + 1\cr
y + 1
} \right ]

The coefficient matrix of this system is nonsingular, hence invertible (Theorem NI), and we can employ its inverse to find a solution (Theorem TTMI, Theorem SNCM),
\left [\array{
{a}_{1}
\cr
{a}_{2} } \right ] ={ \left [\array{
2&7\cr
1&4 } \right ]}^{−1}\left [\array{
x + 1
\cr
y + 1 } \right ] = \left [\array{
4 &−7\cr
−1 & 2
} \right ]\left [\array{
x + 1\cr
y + 1
} \right ] = \left [\array{
4x − 7y − 3\cr
−x + 2y + 1 } \right ]

We could chase through the above implications backwards and take the existence of these solutions as sufficient evidence for R being a spanning set for C. Instead, let us view the above as simply scratchwork and now get serious with a simple direct proof that R is a spanning set. Ready? Suppose (x,\kern 1.95872pt y) is any vector from C, then compute the following linear combination using the definitions of the operations in C,
This final sequence of computations in C is sufficient to demonstrate that any element of C can be written (or expressed) as a linear combination of the two vectors in R, so C ⊆\left \langle R\right \rangle . Since the reverse inclusion \left \langle R\right \rangle ⊆ C is trivially true, C = \left \langle R\right \rangle and we say R spans C (Definition TSVS). Notice that this demonstration is no more or less valid if we hide from the reader our scratchwork that suggested {a}_{1} = 4x − 7y − 3 and {a}_{2} = −x + 2y + 1. ⊠
In Chapter R we will take up the matter of representations fully, where Theorem VRRB will be critical for Definition VR. We will now motivate and prove a critical theorem that tells us how to “represent” a vector. This theorem could wait, but working with it now will provide some extra insight into the nature of linearly independent spanning sets. First an example, then the theorem.
Example AVR
A vector representation
Consider the set
S = \left \{\left [\array{
−7\cr
5
\cr
1 } \right ],\kern 1.95872pt \left [\array{
−6\cr
5
\cr
0 } \right ],\kern 1.95872pt \left [\array{
−12\cr
7
\cr
4 } \right ]\right \}

from the vector space {ℂ}^{3}. Let A be the matrix whose columns are the set S, and verify that A is nonsingular. By Theorem NMLIC the elements of S form a linearly independent set. Suppose that b ∈ {ℂ}^{3}. Then ℒS\kern 1.95872pt \left (A,\kern 1.95872pt b\right ) has a (unique) solution (Theorem NMUS) and hence is consistent. By Theorem SLSLC, b ∈\left \langle S\right \rangle . Since b is arbitrary, this is enough to show that \left \langle S\right \rangle = {ℂ}^{3}, and therefore S is a spanning set for {ℂ}^{3} (Definition TSVS). (This set comes from the columns of the coefficient matrix of Archetype B.)
Now examine the situation for a particular choice of b, say b = \left [\array{ −33\cr 24 \cr 5 } \right ]. Because S is a spanning set for {ℂ}^{3}, we know we can write b as a linear combination of the vectors in S,
\left [\array{
−33\cr
24
\cr
5 } \right ] = (−3)\left [\array{
−7\cr
5
\cr
1 } \right ]+(5)\left [\array{
−6\cr
5
\cr
0 } \right ]+(2)\left [\array{
−12\cr
7
\cr
4 } \right ].

The nonsingularity of the matrix A tells that the scalars in this linear combination are unique. More precisely, it is the linear independence of S that provides the uniqueness. We will refer to the scalars {a}_{1} = −3, {a}_{2} = 5, {a}_{3} = 2 as a “representation of b relative to S.” In other words, once we settle on S as a linearly independent set that spans {ℂ}^{3}, the vector b is recoverable just by knowing the scalars {a}_{1} = −3, {a}_{2} = 5, {a}_{3} = 2 (use these scalars in a linear combination of the vectors in S). This is all an illustration of the following important theorem, which we prove in the setting of a general vector space. ⊠
Theorem VRRB
Vector Representation Relative to a Basis
Suppose that V is a vector
space and B = \left \{{v}_{1},\kern 1.95872pt {v}_{2},\kern 1.95872pt {v}_{3},\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {v}_{m}\right \} is a linearly
independent set that spans V .
Let w be any vector
in V . Then there
exist unique scalars {a}_{1},\kern 1.95872pt {a}_{2},\kern 1.95872pt {a}_{3},\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {a}_{m}
such that
w = {a}_{1}{v}_{1} + {a}_{2}{v}_{2} + {a}_{3}{v}_{3} + \mathrel{⋯} + {a}_{m}{v}_{m}.

Proof That w can be written as a linear combination of the vectors in B follows from the spanning property of the set (Definition TSVS). This is good, but not the meat of this theorem. We now know that for any choice of the vector w there exist some scalars that will create w as a linear combination of the basis vectors. The real question is: Is there more than one way to write w as a linear combination of \{{v}_{1},\kern 1.95872pt {v}_{2},\kern 1.95872pt {v}_{3},\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {v}_{m}\}? Are the scalars {a}_{1},\kern 1.95872pt {a}_{2},\kern 1.95872pt {a}_{3},\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {a}_{m} unique? (Technique U)
Assume there are two ways to express w as a linear combination of \{{v}_{1},\kern 1.95872pt {v}_{2},\kern 1.95872pt {v}_{3},\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {v}_{m}\}. In other words there exist scalars {a}_{1},\kern 1.95872pt {a}_{2},\kern 1.95872pt {a}_{3},\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {a}_{m} and {b}_{1},\kern 1.95872pt {b}_{2},\kern 1.95872pt {b}_{3},\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {b}_{m} so that
Then notice that
But this is a relation of linear dependence on a linearly independent set of vectors (Definition RLD)! Now we are using the other assumption about B, that \{{v}_{1},\kern 1.95872pt {v}_{2},\kern 1.95872pt {v}_{3},\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {v}_{m}\} is a linearly independent set. So by Definition LI it must happen that the scalars are all zero. That is,
And so we find that the scalars are unique. ■
This is a very typical use of the hypothesis that a set is linearly independent — obtain a relation of linear dependence and then conclude that the scalars must all be zero. The result of this theorem tells us that we can write any vector in a vector space as a linear combination of the vectors in a linearly independent spanning set, but only just. There is only enough raw material in the spanning set to write each vector one way as a linear combination. So in this sense, we could call a linearly independent spanning set a “minimal spanning set.” These sets are so important that we will give them a simpler name (“basis”) and explore their properties further in the next section.
\left \{\left [\array{
1 &3\cr
−2 &4} \right ],\kern 1.95872pt \left [\array{
−2& 3\cr
3 &−5
} \right ],\kern 1.95872pt \left [\array{
0 &9\cr
−1 &3} \right ]\right \}

S = \left \{\left [\array{
6\cr
2
\cr
1} \right ],\kern 1.95872pt \left [\array{
4\cr
−3
\cr
1} \right ],\kern 1.95872pt \left [\array{
5\cr
8
\cr
2} \right ]\right \}

is linearly independent and spans {ℂ}^{3}. Write the vector x = \left [\array{ −6\cr 2 \cr 2} \right ] a linear combination of the elements of S. How many ways are there to answer this question, and which theorem allows you to say so?
C20 In the vector space of 2 × 2 matrices, {M}_{22}, determine if the set S below is linearly independent.
S = \left \{\left [\array{
2&−1\cr
1& 3 } \right ],\kern 1.95872pt \left [\array{
0 &4\cr
−1 &2 } \right ],\kern 1.95872pt \left [\array{
4&2\cr
1&3 } \right ]\right \}

Contributed by Robert Beezer Solution [974]
C21 In the crazy vector space C
(Example CVS), is the set S = \left \{(0,\kern 1.95872pt 2),\ (2,\kern 1.95872pt 8)\right \}
linearly independent?
Contributed by Robert Beezer Solution [975]
C22 In the vector space of polynomials {P}_{3}, determine if the set S is linearly independent or linearly dependent.
S = \left \{2 + x − 3{x}^{2} − 8{x}^{3},\kern 1.95872pt 1 + x + {x}^{2} + 5{x}^{3},\kern 1.95872pt 3 − 4{x}^{2} − 7{x}^{3}\right \}

Contributed by Robert Beezer Solution [977]
C23 Determine if the set S = \left \{(3,\kern 1.95872pt 1),\kern 1.95872pt (7,\kern 1.95872pt 3)\right \}
is linearly independent in the crazy vector space
C
(Example CVS).
Contributed by Robert Beezer Solution [978]
C24 In the vector space of realvalued functions F = \left \{f\mathrel{∣}f : {ℝ}^{} → {ℝ}^{}\right \}, determine if the following set S is linearly independent.
S = \left \{{\mathop{sin}\nolimits }^{2}x,{\mathop{ cos}\nolimits }^{2}x, 2\right \}

Contributed by Chris Black Solution [979]
C25 Let
S = \left \{\left [\array{
1&2\cr
2&1 } \right ],\left [\array{
2 &1\cr
−1 &2 } \right ],\left [\array{
0&1\cr
1&2 } \right ]\right \}

Contributed by Chris Black Solution [979]
C26 Let
S = \left \{\left [\array{
1&2\cr
2&1 } \right ],\left [\array{
2 &1\cr
−1 &2 } \right ],\left [\array{
0&1\cr
1&2 } \right ],\left [\array{
1&0\cr
1&1 } \right ],\left [\array{
1&4\cr
0&3 } \right ]\right \}

Contributed by Chris Black Solution [982]
C30 In Example LIM32, find another nontrivial relation of linear dependence on the linearly
dependent set of 3 × 2
matrices, S.
Contributed by Robert Beezer
C40 Determine if the set T = \left \{{x}^{2} − x + 5,\kern 1.95872pt 4{x}^{3} − {x}^{2} + 5x,\kern 1.95872pt 3x + 2\right \}
spans the vector space of polynomials with degree 4 or less,
{P}_{4}.
Contributed by Robert Beezer Solution [986]
C41 The set W is a subspace of {M}_{22}, the vector space of all 2 × 2 matrices. Prove that S is a spanning set for W.
Contributed by Robert Beezer Solution [987]
C42 Determine if the set S = \left \{(3,\kern 1.95872pt 1),\kern 1.95872pt (7,\kern 1.95872pt 3)\right \}
spans the crazy vector space C
(Example CVS).
Contributed by Robert Beezer Solution [988]
M10 Halfway through Example SSP4, we need to show that the system of equations
ℒS\kern 1.95872pt \left (\left [\array{
0 & 0 & 0 & 1\cr
0 & 0 & 1 & −8
\cr
0 & 1 &−6& 24\cr
1 &−4 & 12 &−32
\cr
−2& 4 &−8& 16\cr
} \right ],\kern 1.95872pt \left [\array{
a\cr
b
\cr
c\cr
d
\cr
e } \right ]\right )

is consistent for every choice of the vector of constants satisfying 16a + 8b + 4c + 2d + e = 0.
Express the column space of the coefficient matrix of this system as a null
space, using Theorem FS. From this use Theorem CSCS to establish that
the system is always consistent. Notice that this approach removes from
Example SSP4 the need to rowreduce a symbolic matrix.
Contributed by Robert Beezer Solution [990]
T40 Prove the following variation of Theorem EMMVP: Suppose that
B = \left \{{u}_{1},\kern 1.95872pt {u}_{2},\kern 1.95872pt {u}_{3},\kern 1.95872pt \mathop{\mathop{…}},\kern 1.95872pt {u}_{n}\right \} is a basis for
{ℂ}^{n}. Suppose
also that A
and B are
m × n matrices
such that A{u}_{i} = B{u}_{i}
for every 1 ≤ i ≤ n.
Then A = B.
Can you modify the hypothesis further and obtain a generalization of
Theorem EMMVP?
Contributed by Robert Beezer
T50 Suppose that V
is a vector space and u,\kern 1.95872pt v ∈ V
are two vectors in V .
Use the definition of linear independence to prove that
S = \left \{u,\kern 1.95872pt v\right \} is a
linearly dependent set if and only if one of the two vectors is a scalar multiple of
the other. Prove this directly in the context of an abstract vector space
(V ),
without simply giving an upgraded version of Theorem DLDS for the special case
of just two vectors.
Contributed by Robert Beezer Solution [990]
C20 Contributed by Robert Beezer Statement [968]
Begin with a relation of linear dependence on the vectors in
S and
massage it according to the definitions of vector addition and scalar multiplication
in {M}_{22},
By our definition of matrix equality (Definition ME) we arrive at a homogeneous system of linear equations,
The coefficient matrix of this system rowreduces to the matrix,
\left [\array{
\text{1}&0&0\cr
0&\text{1 } &0
\cr
0&0&\text{1}\cr
0&0 &0} \right ]

and from this we conclude that the only solution is {a}_{1} = {a}_{2} = {a}_{3} = 0. Since the relation of linear dependence (Definition RLD) is trivial, the set S is linearly independent (Definition LI).
C21 Contributed by Robert Beezer Statement [968]
We begin with a relation of linear dependence using unknown scalars
a and
b. We
wish to know if these scalars must both be zero. Recall that the zero vector in
C is
(−1,\kern 1.95872pt − 1) and
that the definitions of vector addition and scalar multiplication are not what we
might expect.
From this we obtain two equalities, which can be converted to a homogeneous system of equations,
This homogeneous system has a singular coefficient matrix (Theorem SMZD), and so has more than just the trivial solution (Definition NM). Any nontrivial solution will give us a nontrivial relation of linear dependence on S. So S is linearly dependent (Definition LI).
C22 Contributed by Robert Beezer Statement [968]
Begin with a relation of linear dependence (Definition RLD),
{a}_{1}\left (2 + x − 3{x}^{2} − 8{x}^{3}\right ) + {a}_{
2}\left (1 + x + {x}^{2} + 5{x}^{3}\right ) + {a}_{
3}\left (3 − 4{x}^{2} − 7{x}^{3}\right ) = 0

Massage according to the definitions of scalar multiplication and vector addition in the definition of {P}_{3} (Example VSP) and use the zero vector dro this vector space,
\left (2{a}_{1} + {a}_{2} + 3{a}_{3}\right )+\left ({a}_{1} + {a}_{2}\right )x+\left (−3{a}_{1} + {a}_{2} − 4{a}_{3}\right ){x}^{2}+\left (−8{a}_{
1} + 5{a}_{2} − 7{a}_{3}\right ){x}^{3} = 0+0x+0{x}^{2}+0{x}^{3}

The definition of the equality of polynomials allows us to deduce the following four equations,
Rowreducing the coefficient matrix of this homogeneous system leads to the unique solution {a}_{1} = {a}_{2} = {a}_{3} = 0. So the only relation of linear dependence on S is the trivial one, and this is linear independence for S (Definition LI).
C23 Contributed by Robert Beezer Statement [969]
Notice, or discover, that the following gives a nontrivial relation of linear dependence
on S in
C, so by Definition LI,
the set S
is linearly dependent.
2(3,\kern 1.95872pt 1) + (−1)(7,\kern 1.95872pt 3) = (7,\kern 1.95872pt 3) + (−9,\kern 1.95872pt − 5) = (−1,\kern 1.95872pt − 1) = 0

C24 Contributed by Chris Black Statement [969]
One of the fundamental identities of trigonometry is
{\mathop{ sin}\nolimits }^{2}(x) +{\mathop{ cos}\nolimits }^{2}(x) = 1. Thus, we have a
dependence relation 2({\mathop{sin}\nolimits }^{2}x) + 2({\mathop{cos}\nolimits }^{2}x) + (−1)(2) = 0,
and the set is linearly dependent.
C25 Contributed by Chris Black Statement [969]
Applying Definition ME, this leads to the linear system
We need to rowreduce the augmented matrix of this system by hand due to the symbols x, y, z, and w in the vector of constants.
With the apperance of a leading 1 possible in the last column, by Theorem RCLS there will exist some matrices B = \left [\array{ x&y\cr z &w } \right ] so that the linear system above has no solution (namely, whenever 5y − 3x − 3z − w\mathrel{≠}0), so the set S does not span {M}_{2,2}. (For example, you can verify that there is no solution when B = \left [\array{ 3&3\cr 3&2} \right ].)
This requires the same work that was done in part (a), with x = y = z = w = 0. In that case, the coefficient matrix rowreduces to have a leading 1 in each of the first three columns and a row of zeros on the bottom, so we know that the only solution to the matrix equation is α = β = γ = 0. So the set S is linearly independent.
C26 Contributed by Chris Black Statement [970]
Thus, we have
This system will have a solution for every vector on the right side if the rowreduced coefficient matrix has a leading one in every row, since then it is never possible to have a leading 1 appear in teh final colum of a rowreduced augmented matrix.
Since there is a leading one in each row of the rowreduced coefficient matrix, there is a solution for every vector \left [\array{ x\cr y \cr z\cr w } \right ], which means that there is a solution to the original equation for every matrix \left [\array{ x&y\cr z &w } \right ]. Thus, the original four matrices span {M}_{2,2}.
is a = b = c = d = e = 0.
We have
so we need to find the nullspace of the matrix
We rowreduced this matrix in part (a), and found that there is a column without a leading 1, which correspons to a free variable in a description of the solution set to the homogeneous system, so the nullspace is nontrivial and there are an infinite number of solutions to
Thus, this set of matrices is not linearly independent.
C40 Contributed by Robert Beezer Statement [971]
The polynomial {x}^{4}
is an element of {P}_{4}.
Can we write this element as a linear combination of the elements of
T? To wit, are
there scalars {a}_{1},
{a}_{2},
{a}_{3} such
that
Massaging the right side of this equation, according to the definitions of Example VSP, and then equating coefficients, leads to an inconsistent system of equations (check this!). As such, T is not a spanning set for {P}_{4}.
C41 Contributed by Robert Beezer Statement [971]
We want to show that W = \left \langle S\right \rangle
(Definition TSVS), which is an equality of sets (Definition SE).
First, show that \left \langle S\right \rangle ⊆ W. Begin by checking that each of the three matrices in S is a member of the set W. Then, since W is a vector space, the closure properties (Property AC, Property SC) guarantee that every linear combination of elements of S remains in W.
Second, show that W ⊆\left \langle S\right \rangle . We want to convince ourselves that an arbitrary element of W is a linear combination of elements of S. Choose
x = \left [\array{
a&b\cr
c&d } \right ] ∈ W

The values of a,\kern 1.95872pt b,\kern 1.95872pt c,\kern 1.95872pt d are not totally arbitrary, since membership in W requires that 2a − 3b + 4c − d = 0. Now, rewrite as follows,
C42 Contributed by Robert Beezer Statement [971]
We will try to show that S
spans C. Let
(x,\kern 1.95872pt y) be an arbitrary
element of C and
search for scalars {a}_{1}
and {a}_{2}
such that
Equality in C leads to the system
This system has a singular coefficient matrix whose column space is simply \left \langle \left [\array{ 2\cr 1 } \right ]\right \rangle . So any choice of x and y that causes the column vector \left [\array{ x + 1\cr y + 1 } \right ] to lie outside the column space will lead to an inconsistent system, and hence create an element (x,\kern 1.95872pt y) that is not in the span of S. So S does not span C.
For example, choose x = 0 and y = 5, and then we can see that \left [\array{ 1\cr 6 } \right ]∉\left \langle \left [\array{ 2\cr 1 } \right ]\right \rangle and we know that (0,\kern 1.95872pt 5) cannot be written as a linear combination of the vectors in S. A shorter solution might begin by asserting that (0,\kern 1.95872pt 5) is not in \left \langle S\right \rangle and then establishing this claim alone.
M10 Contributed by Robert Beezer Statement [972]
Theorem FS provides the matrix
L = \left [\array{
\text{1}&{1\over
2}&{1\over
4}&{1\over
8}& {1\over _
16} } \right ]

and so if A denotes the coefficient matrix of the system, then C\kern 1.95872pt \left (A\right ) = N\kern 1.95872pt \left (L\right ). The single homogeneous equation in ℒS\kern 1.95872pt \left (L,\kern 1.95872pt 0\right ) is equivalent to the condition on the vector of constants (use a,\kern 1.95872pt b,\kern 1.95872pt c,\kern 1.95872pt d,\kern 1.95872pt e as variables and then multiply by 16).
T50 Contributed by Robert Beezer Statement [972]
( ⇒) If S is linearly dependent, then there are scalars α and β, not both zero, such that αu + βv = 0. Suppose that α\mathrel{≠}0, the proof proceeds similarly if β\mathrel{≠}0. Now,
which shows that u is a scalar multiple of v.
( ⇐) Suppose now that u is a scalar multiple of v. More precisely, suppose there is a scalar γ such that u = γv. Then
This is a relation of linear of linear dependence on S (Definition RLD), which is nontrivial since one of the scalars is − 1. Therefore S is linearly dependent by Definition LI.
Be careful using this theorem. It is only applicable to sets of two vectors. In particular, linear dependence in a set of three or more vectors can be more complicated than just one vector being a scalar multiple of another.