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...

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.

Asked on by bobby9901

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to consider the matrix `A_(m x n)`  such that:

`A = ((a_(11),.....,a_(1n)),(a_(21),....,a_(2n)),(....),(a_(m1),....,a_(mn)))`

The problem provides the information that bar x is the solution for matrix equation `A*bar x = bar b.`

You need to perform the matrix multiplication, such that:

`((a_(11),.....,a_(1n)),(a_(21),....,a_(2n)),(....),((a_(m1),....,a_(mn))))*((bar x_1),(bar x_2),(...),(bar x_n)) = ((a_(11)*bar x_1 + ...a_(1n)*bar x_n),(a_(21)*barx_1 + ... + a_(2n)*bar x_n),(...),(a_(m1)*bar x_1 + ...+a_(mn)*bar x_n))`

Hence, performing the multiplication of given matrices yields the column vector `bar b ` = `((a_(11)*bar x_1 + ...a_(1n)*bar x_n),(a_(21)*barx_1 + ... + a_(2n)*bar x_n),(...),(a_(m1)*bar x_1 + ...+a_(mn)*bar x_n))`

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