The coefficient matrix A = [1 -2 -3 3 1 -2 5 7 -5 -4 3 -5 -8 11 0 -1 3 4 -3 -2]

Question

The coefficient matrix A = [1 -2 -3 3 1 -2 5 7 -5 -4 3 -5 -8 11 0 -1 3 4 -3 -2] row reduces to U = [1 0 -1 0 2 0 1 1 0 -1 0 0 0 1 -1 0 0 0 0 0] true or false: the column space of A is identical to the column space of U what is the rank of the matrix A ? find linearly independent vectors which span the column space of A true or false: the row space of A is identical to the row space of U find linearly independent vectors which span the row space of A true or false: the null space of A is identical to the null space of U find linearly independent vectors which span the null space of A

Explanation / Answer

Given A=

RREF(A)=U=

First of all we need to know that to findout column, row and null space of a matrix

first we need to find the RREF of the matrix

(a)It is false that the column space of A is identical to the column space of U

Because U is the RREF of A and we ccanot reduce U so, that the pivot elements

in U are itself forms the column space of U whicha re not identical with the column

space of U.

(b)rank:

In the RREF(A)=U The number of non zero rows are 3.

so, the rank (A)=3.

(c)In the RREF(A)=U the first second and fourth columns have pivot element 1.

so, the corresponding 1,2&4 column vectors in A are the linearly inependent vectors.

which spans the column space of A.

C(A)=spans{

,

,

(d)It is true that the row space of A is identical to the row space of U

Because U is the RREF of A

(e)In the RREF(A)=U

thus the linearly inependent vectors.which spans the row space of A.

R(A)=spans

1 -2 -3 3 1 -2 5 7 -5 -4 3 -5 -8 11 0 -1 3 4 -3 -2

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