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1. If Ais an n-by-m matrix, then the linear transformation x↦Axis a linear transformation from Rmto Rn. Saying that it is onto Rnis then the same as saying that the columns of Aspan Rn, and is also the same as saying that an echelon form of Ahas a pivot in every row.2)the linear transformation x↦Axassociated to a matrix is injective if and only if the columns of Aare linearly independent.
3)the linear transformation x↦Axassociated to a matrix is surjective if and only if