Archetype U

From A First Course in Linear Algebra
Version 1.04
© 2004.
Licensed under the GNU Free Documentation License.
http://linear.ups.edu/

Summary Domain is matrices, codomain is column vectors. Domain has dimension 6, while codomain has dimension 4. Can’t be injective, is surjective. A linear transformation: (Definition LT)

T : M_{23} \mapsto ℂ^{4}, T ([\begin{matrix} a & b & c \\ d & e & f \end{matrix}]) = [\begin{matrix} a + 2 b + 12 c - 3 d + e + 6 f \\ 2 a - b - c + d - 11 f \\ a + b + 7 c + 2 d + e - 3 f \\ a + 2 b + 12 c + 5 e - 5 f \end{matrix}]

A basis for the null space of the linear transformation: (Definition KLT)

\{[\begin{matrix} 3 & - 4 & 0 \\ 1 & 2 & 1 \end{matrix}], [\begin{matrix} - 2 & - 5 & 1 \\ 0 & 0 & 0 \end{matrix}]\}

Injective: No. (Definition ILT)
Since the kernel is nontrivial Theorem KILT tells us that the linear transformation is not injective. Also, since the rank can not exceed 4, we are guaranteed to have a nullity of at least 2, just from checking dimensions of the domain and the codomain. In particular, verify that

\begin{array}{l} T ([\begin{matrix} 1 & 10 & - 2 \\ 3 & - 1 & 1 \end{matrix}]) & = [\begin{matrix} - 7 \\ - 14 \\ - 1 \\ - 13 \end{matrix}] & T ([\begin{matrix} 5 & - 3 & - 1 \\ 5 & 3 & 3 \end{matrix}]) & = [\begin{matrix} - 7 \\ - 14 \\ - 1 \\ - 13 \end{matrix}] \end{array}

This demonstration that $T$ is not injective is constructed with the observation that

\begin{array}{l} [\begin{matrix} 5 & - 3 & - 1 \\ 5 & 3 & 3 \end{matrix}] & = [\begin{matrix} 1 & 10 & - 2 \\ 3 & - 1 & 1 \end{matrix}] + [\begin{matrix} 4 & - 13 & 1 \\ 2 & 4 & 2 \end{matrix}] \\ and \\ z & = [\begin{matrix} 4 & - 13 & 1 \\ 2 & 4 & 2 \end{matrix}] \in K (T) \end{array}

so the vector $z$ effectively “does nothing” in the evaluation of $T$ .

A basis for the range of the linear transformation: (Definition RLT)
Evaluate the linear transformation on a standard basis to get a spanning set for the range (Theorem SSRLT):

[\begin{matrix} 1 \\ 2 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 2 \\ - 1 \\ 1 \\ 2 \end{matrix}], [\begin{matrix} 12 \\ - 1 \\ 7 \\ 12 \end{matrix}], [\begin{matrix} - 3 \\ 1 \\ 2 \\ 0 \end{matrix}], [\begin{matrix} 1 \\ 0 \\ 1 \\ 5 \end{matrix}], [\begin{matrix} 6 \\ - 11 \\ - 3 \\ - 5 \end{matrix}]

If the linear transformation is injective, then the set above is guaranteed to be linearly independent (Theorem ILTLI). This spanning set may be converted to a “nice” basis, by making the vectors the rows of a matrix (perhaps after using a vector reperesentation), row-reducing, and retaining the nonzero rows (Theorem BRS), and perhaps un-coordinatizing. A basis for the range is:

\{[\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix}]\}

Surjective: Yes. (Definition SLT)
A basis for the range is the standard basis of $ℂ^{4}$ , so $ℛ (T) = ℂ^{4}$ and Theorem RSLT tells us $T$ is surjective. Or, the dimension of the range is 4, and the codomain ( $ℂ^{4}$ ) has dimension 4. So the transformation is surjective.

Subspace dimensions associated with the linear transformation. Examine parallels with earlier results for matrices. Verify Theorem RPNDD.

\begin{array}{l} Domain dimension: 6 & Rank: 4 & Nullity: 2 \end{array}

Invertible: No.
The relative sizes of the domain and codomain mean the linear transformation cannot be injective. (Theorem ILTIS)

Matrix representation (Definition MR):

\begin{array}{l} B & = \{[\begin{matrix} 1 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}], [\begin{matrix} 0 & 1 & 0 \\ 0 & 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 & 1 \\ 0 & 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 & 0 \\ 1 & 0 & 0 \end{matrix}], [\begin{matrix} 0 & 0 & 0 \\ 0 & 1 & 0 \end{matrix}], [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 1 \end{matrix}]\} \\ C & = \{[\begin{matrix} 1 \\ 0 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 1 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 0 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 0 \\ 0 \\ 0 \\ 1 \end{matrix}]\} \\ M_{B, C}^{T} & = [\begin{matrix} 1 & 2 & 12 & - 3 & 1 & 6 \\ 2 & - 1 & - 1 & 1 & 0 & - 11 \\ 1 & 1 & 7 & 2 & 1 & - 3 \\ 1 & 2 & 12 & 0 & 5 & - 5 \end{matrix}] \end{array}

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