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

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hockeyfan54 | Student | Salutatorian

Posted April 16, 2013 at 2:22 AM via web

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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 `Ax=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.` 

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tiburtius | High School Teacher | (Level 3) Associate Educator

Posted April 16, 2013 at 5:31 AM (Answer #1)

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Let `A_1,A_2,ldots,A_n` be columns of matrix `A.` Now we have

`[(A_1,A_2,ldots,A_n)][(a_1),(a_2),(.),(.),(.),(a_n)]=b`

Hence we can write vector `b` as 

`b=a_1A_1+a_2A_2+cdots+a_nA_n`

which is a linear combination of columns of `A`. Thus `b` is in the column space of  `A.`

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