I'm trying to learn column space currently and I'm confused about the meaning of rows and columns.(adsbygoogle = window.adsbygoogle || []).push({});

So I'm given this definition for column space:

"The column space of matrix A is the set Col A of all linear combinations of the columns of A"

Given the matrix A:

[ 1 -3 -4 ]

[ -4 6 -2 ]

[ -3 7 6 ]

b=

[ 3 ]

[ 3 ]

[ -4 ]

Determine if b is in the column space of A.

My books solves by row reducing [ A b ].

Has this always been what I was solving for whenever I row reduced an augmented matrix to obtain x for Ax = b?

For example, when I'm given a system of linear equation such as:

2x_{1}+ 3x_{2}= 5

1x_{1}+ 2x_{2}= 3

and I have to solve forx.

Do the columns of the coefficient matrix of this system of linear equation, have the same meaning as the matrix above, vectors?

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# Matrices: Rows and Columns Meaning

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