math notes - 1 If A is an n-by-m matrix then the linear...

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1. If A is an n-by-m matrix, then the linear transformation x Ax is a linear transformation from R m to R n . Saying that it is onto R n is then the same as saying that the columns of A span R n , and is also the same as saying that an echelon form of A has a pivot in every row. 2) the linear transformation x Ax associated to a matrix is injective if and only if the columns of A are linearly independent.
3) the linear transformation x Ax associated to a matrix is surjective if and only if

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