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Question
Any help would be greatly appreciated. I a having difficulty understanding this problem.

Take an m by n matrix A. Let vector b be a vector in R^m. Assume that

vector x = <a_1,...,a_n>

is a solution for the matrix equation A(vector x) = vector b. Show that vector b is in the column space of A by writing vector b as a linear combination of the columns of A.

Answer
For A an mxn matrix and x an nx1 vector, the product Ax should be an mx1 vector, by the rules of matrix multiplication. For A, the mxn form means m rows and n columns, so the n columns of A represent vectors with m components. Therefore we can write

[ a11  a12 ... a1n   ] {x1]   [b1]
[ a21  a22 ... a2n   ] [x2]= [b2]
[  .          ]   .         .
         [xn]

[ am1 am2 ... amn ]          [bm]

or

[ a11]        [a12]       ... [a1n]        = [b1]
[ a21][x1]  [a22][x2] ... [a2n][xn  ]= [b2]
[  .    ]        [  .  ]          [  .   ]       = [  .  ]
[ am1]          .          [amn       = [bm].

where the components of the vector x serve as coefficients for the column vectors of A so give a linear combination for b.

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