I'm trying to learn column space currently and I'm confused about the meaning of rows and columns. 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: 2x1 + 3x2 = 5 1x1 + 2x2 = 3 and I have to solve for x. Do the columns of the coefficient matrix of this system of linear equation, have the same meaning as the matrix above, vectors?