Archetype J   

From A First Course in Linear Algebra
Version 2.00
© 2004.
Licensed under the GNU Free Documentation License.
http://linear.ups.edu/

Summary  System with six equations, nine variables. Consistent. Null space of coefficient matrix has dimension 5.

A system of linear equations (Definition SLE):

x1 + 2x2 2x3 + 9x4 + 3x5 5x6 2x7 + x8 + 27x9 = 5 2x1 + 4x2 + 3x3 + 4x4 x5 + 4x6 + 10x7 + 2x8 23x9 = 18 x1 + 2x2 + x3 + 3x4 + x5 + x6 + 5x7 + 2x8 7x9 = 6 2x1 + 4x2 + 3x3 + 4x4 7x5 + 2x6 + 4x7 11x9 = 20 x1 + 2x2 + 5x4 + 2x5 4x6 + 3x7 + 8x8 + 13x9 = 4 3x1 6x2 x3 13x4 + 2x5 5x6 4x7 + 13x8 + 10x9 = 29

Some solutions to the system of linear equations (not necessarily exhaustive):
x1 = 6, x2 = 0, x3 = 1, x4 = 0, x5 = 1, x6 = 2, x7 = 0, x8 = 0, x9 = 0
x1 = 4, x2 = 1, x3 = 1, x4 = 0, x5 = 1, x6 = 2, x7 = 0, x8 = 0, x9 = 0
x1 = 17, x2 = 7, x3 = 3, x4 = 2, x5 = 1, x6 = 14, x7 = 1, x8 = 3, x9 = 2
x1 = 11, x2 = 6, x3 = 1, x4 = 5, x5 = 4, x6 = 7, x7 = 3, x8 = 1, x9 = 1

Augmented matrix of the linear system of equations (Definition AM):

1 2 2 9 3 52 1 27 5 2 4 3 4 1 4 10 2 23 18 1 2 1 3 1 1 5 2 7 6 2 4 3 4 7 2 4 0 11 20 1 2 0 5 2 4 3 8 13 4 36113 2 5413 10 29

Matrix in reduced row-echelon form, row-equivalent to augmented matrix:

120 5 0012 3 6 0 012003 5 61 0 00 0 101 1 11 0 00 0 01023 2 0 00 0 000 0 0 0 0 00 0 000 0 0 0

Analysis of the augmented matrix (Notation RREFA):

r = 4 D = 1,3,5,6 F = 2,4,7,8,9,10

Vector form of the solution set to the system of equations (Theorem VFSLS). Notice the relationship between the free variables and the set F above. Also, notice the pattern of 0’s and 1’s in the entries of the vectors corresponding to elements of the set F for the larger examples.
x1 x2 x3 x4 x5 x6 x7 x8 x9 = 6 0 1 0 1 2 0 0 0 + x2 2 1 0 0 0 0 0 0 0 + x4 5 0 2 1 0 0 0 0 0 + x7 1 0 3 0 1 0 1 0 0 + x8 2 0 5 0 1 2 0 1 0 + x9 3 0 6 0 1 3 0 0 1

Given a system of equations we can always build a new, related, homogeneous system (Definition HS) by converting the constant terms to zeros and retaining the coefficients of the variables. Properties of this new system will have precise relationships with various properties of the original system.

x1 + 2x2 2x3 + 9x4 + 3x5 5x6 2x7 + x8 + 27x9 = 0 2x1 + 4x2 + 3x3 + 4x4 x5 + 4x6 + 10x7 + 2x8 23x9 = 0 x1 + 2x2 + x3 + 3x4 + x5 + x6 + 5x7 + 2x8 7x9 = 0 2x1 + 4x2 + 3x3 + 4x4 7x5 + 2x6 + 4x7 11x9 = 0 x1 + 2x2 + +5x4 + 2x5 4x6 + 3x7 + 8x8 + 13x9 = 0 3x1 6x2 x3 13x4 + 2x5 5x6 4x7 + 13x8 + 10x9 = 0

Some solutions to the associated homogenous system of linear equations (not necessarily exhaustive):
x1 = 0, x2 = 0, x3 = 0, x4 = 0, x5 = 0, x6 = 0, x7 = 0, x8 = 0, x9 = 0
x1 = 2, x2 = 1, x3 = 0, x4 = 0, x5 = 0, x6 = 0, x7 = 0, x8 = 0, x9 = 0
x1 = 23, x2 = 7, x3 = 4, x4 = 2, x5 = 0, x6 = 12, x7 = 1, x8 = 3, x9 = 2
x1 = 17, x2 = 6, x3 = 2, x4 = 5, x5 = 3, x6 = 5, x7 = 3, x8 = 1, x9 = 1

Form the augmented matrix of the homogenous linear system, and use row operations to convert to reduced row-echelon form. Notice how the entries of the final column remain zeros:

120 5 0012 3 0 0 012003 5 60 0 00 0 101 1 10 0 00 0 010230 0 00 0 000 0 0 0 0 00 0 000 0 0 0

Analysis of the augmented matrix for the homogenous system (Notation RREFA). Notice the slight variation for the same analysis of the original system only when the original system was consistent:

r = 4 D = 1,3,5,6 F = 2,4,7,8,9,10

Coefficient matrix of original system of equations, and of associated homogenous system. This matrix will be the subject of further analysis, rather than the systems of equations.

1 2 2 9 3 52 1 27 2 4 3 4 1 4 10 2 23 1 2 1 3 1 1 5 2 7 2 4 3 4 7 2 4 0 11 1 2 0 5 2 4 3 8 13 36113 2 5413 10

Matrix brought to reduced row-echelon form:

120 5 0012 3 0 012003 5 6 0 00 0 101 1 1 0 00 0 01023 0 00 0 000 0 0 0 00 0 000 0 0

Analysis of the row-reduced matrix (Notation RREFA):

r = 4 D = 1,3,5,6 F = 2,4,7,8,9

This is the null space of the matrix. The set of vectors used in the span construction is a linearly independent set of column vectors that spans the null space of the matrix (Theorem SSNS, Theorem BNS). Solve the homogenous system with this matrix as the coefficient matrix and write the solutions in vector form (Theorem VFSLS) to see these vectors arise.
2 1 0 0 0 0 0 0 0 , 5 0 2 1 0 0 0 0 0 , 1 0 3 0 1 0 1 0 0 , 2 0 5 0 1 2 0 1 0 , 3 0 6 0 1 3 0 0 1

Column space of the matrix, expressed as the span of a set of linearly independent vectors that are also columns of the matrix. These columns have indices that form the set D above. (Theorem BCS)
1 2 1 2 1 3 , 2 3 1 3 0 1 , 3 1 1 7 2 2 , 5 4 1 2 4 5

The column space of the matrix, as it arises from the extended echelon form of the matrix. The matrix L is computed as described in Definition EEF. This is followed by the column space described by a set of linearly independent vectors that span the null space of L, computed as according to Theorem FS and Theorem BNS. When r = m, the matrix L has no rows and the column space is all of m.
L = 10 186 131 51 131 188 131 77 131 0 1272 13145 131 58 131 14 131
77 131 14 131 0 0 0 1 , 188 131 58 131 0 0 1 0 , 51 131 45 131 0 1 0 0 , 186 131 272 131 1 0 0 0

Column space of the matrix, expressed as the span of a set of linearly independent vectors. These vectors are computed by row-reducing the transpose of the matrix into reduced row-echelon form, tossing out the zero rows, and writing the remaining nonzero rows as column vectors. By Theorem CSRST and Theorem BRS, and in the style of Example CSROI, this yields a linearly independent set of vectors that span the column space.
1 0 0 0 1 29 7 , 0 1 0 0 11 2 94 7 , 0 0 1 0 10 22 , 0 0 0 1 3 2 3

Row space of the matrix, expressed as a span of a set of linearly independent vectors, obtained from the nonzero rows of the equivalent matrix in reduced row-echelon form. (Theorem BRS)
1 2 0 5 0 0 1 2 3 , 0 0 1 2 0 0 3 5 6 , 0 0 0 0 1 0 1 1 1 , 0 0 0 0 0 1 0 2 3

Subspace dimensions associated with the matrix. (Definition NOM, Definition ROM) Verify Theorem RPNC

 Matrix columns:  9  Rank:  4  Nullity:  5