Section 19

Linear transformations

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T (a u + b v) = a T (u) + b T (v)

A linear transformation preserves linear combinations.

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T (x) = A x

Every matrix defines a linear transformation between compatible spaces.

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ker (T) = {v : T (v) = 0}

Vectors that the transformation collapses to zero.

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Im (T) = {T (v) : v \in V}

Set of attainable outputs.

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(S \circ T) (x) = B A x

Composing transformations corresponds to multiplying matrices in the proper order.

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T^{- 1} (y) = A^{- 1} y

Undoes an invertible matrix transformation.

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[v]_{C} = P_{C \leftarrow B} [v]_{B}

Converts coordinates between two bases.