Basis for The Image of T

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Basis for the Image of T

The pivotal columns of \text{img }T$.

Example: Consider the matrix below, which describes a linear transformation from to .

The pivotal 1’s of the row-reduced matrix are in columns 1, 2, and 5, so columns 1, 2, and 5 of the original matrix are a basis for the image. We can express any vector in the image of uniquely as a linear combination of those three vectors. For instance, , can be written

Note that each vector in the basis for the image has four entries, as it must, since the image is a subspace of . But the image is not of course ; a basis for must have four elements.