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The image space of a linear transformation T:V → W is the set of vectors such that

\text{Im(T)} = \{ A \vec{v} |A \vec{v} \in W \}

Given the n×n transformation matrix A, the image space of the transformation will be identical to the column space of A. Therefore, because of the rank–nullity theorem,

n = \text{dim(im(A))} +  \text{dim(null(A))}

The image space of an invertable (that is, the determinant is not zero) n×n matrix is Rn.

See also

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