Mathematical formulas

Algebra

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Algebra
Linear transformations

Section 19

Linear transformations

intermedioviz

T(au+bv)=aT(u)+bT(v)

A linear transformation preserves linear combinations.

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Matrix transformation

intermedioviz

T(x)=Ax

Every matrix defines a linear transformation between compatible spaces.

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intermedioviz

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

Vectors that the transformation collapses to zero.

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intermedioviz

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

Set of attainable outputs.

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Linear composition

intermedioviz

(S\circ T)(x)=BAx

Composing transformations corresponds to multiplying matrices in the proper order.

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Inverse transformation

intermedioviz

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

Undoes an invertible matrix transformation.

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Change of basis

avanzadoviz

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

Converts coordinates between two bases.

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