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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.

If vector b is in R^m, then vector b = <b_1, ..., b_m>.

I will write that as b[m].

It is given that vector x = <a_1, ..., a_n>.

To keep it as x and not confuse it with A, I will make it x[n] = <x_1, ..., x_n>,

The matrix A is defined to be m x n, so write matrix A as A[m,n].

From this, we have A[m,n] * x[n] = b[m].

Take the ith row of matrix A as A[i].

From this, it can be seen that A[i,j] = A[i]*b[j].

Using this notation, it can be seen that the result of the multiplication is given as a linear combination of the rows of A times the elements of x.

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