Linear Algebra Example Problems – Onto Linear Transformations

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An onto linear transformation can “reach” every element in its codomain. More specifically, consider the linear transformation T: Rn to Rm. The linear transformation T is onto if for each b in Rm, there exists an x in Rn such that T(x) = b.

When a linear transformation is described in term of a matrix it is easy to determine if the linear transformation is onto or not by checking the span of the columns of the matrix. If the columns span Rm, then the linear transformation is onto. If the columns do not span Rm, then the linear transformation is not onto.

This video works two different examples. One linear transformation is found to be onto while the other linear transformation is NOT onto.

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