Given the following matrix A: A = Matrix 6 2 6 4 1 7 1 0 3 3 4 7 7 0 9 7 4 2 0 8

Question

Given the following matrix A:
A =
Matrix 6 2 6 4 1 7
1 0 3 3 4 7
7 0 9 7 4 2
0 8 0 7 6 6
1 8 3 10 10 13

and its row reduced echelon form:
rref(A) =
Matrix 1 0 0 0 -13/7 -31/4
0 1 0 0 1/2 6
0 0 1 0 5/3 131/12
0 0 0 1 2/7 -6
0 0 0 0 0 0

(a) Find a basis for the row space of A.
(b) Find a basis for the column space of A.
(c) Find a basis for the null space of A.
(d) What is the rank of this matrix A?
(e) What is the nullity of this matrix A?
(f) Find a basis for the null space of A ^T.
(g) In a smart way, determine a basis for the column space A^ T.

Explanation / Answer

a) row space [ 1, 0, 0, 0, -13/7, -31/4] [ 0, 1, 0, 0, 1/2, 6] [ 0, 0, 1, 0, 5/3, 131/12] [ 0, 0, 0, 1, 2/7, -6] [ 0, 0, 0, 0, 0, 0] b) [ 13/7, 31/4] [ -1/2, -6] [ -5/3, -131/12] [ -2/7, 6] [ 1, 0] [ 0, 1] c) column space [ 1, 0, 0, 0] [ 0, 1, 0, 0] [ 0, 0, 1, 0] [ 0, 0, 0, 1] [ 0, 0, 0, 0] d) rank = 4 e)nullity of matrix 1.0e-15 * -0.5551 -0.5551 -0.3331 -0.1110 -0.4441 0.1110 0.2220 0.1110 0 0 f)basis of null space of aT 0 0 0 0 1 g) column spae is nothing but the columns of the transpose of A [ 1, 0, 0, 0] [ 0, 1, 0, 0] [ 0, 0, 1, 0] [ 0, 0, 0, 1] [ -13/7, 1/2, 5/3, 2/7] [ -31/4, 6, 131/12, -6]

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