Associated Matrix of Composed Transformations

Topics: Algebra - Linear Transformation

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(theorem)

Let $U,V,W$ be vector spaces on $\mathbb{K}$ where $\dim (V)=s$, $\dim (V)=n$, $\dim (W)=t$.

Let:

• $\beta =\{u_1,\dots ,u_s\}$ be a basis of $U$
• $\gamma =\{v_1,\dots ,v_n\}$ be a basis of $V$
• $\delta =\{w_1,\dots ,w_t\}$ be a basis of $W$

…and let $S:U\to V$ and $T:V\to W$ be linear transformations.

Then, the associated matrix of $T\circ S$ (composition) is:

$$[T\circ S{]}_{\beta }^{\delta }=[T{]}_{\gamma }^{\delta }[S{]}_{\beta }^{\gamma }$$

Inverse Transformations and Isomorphism §

(observation)

Let $T$ be an Isomorphism.

• $T:V\to W$ is linear; $T^{-1}:W\to V$ is also linear.
• Let $\beta$ be a basis of $V$
• Let ${\beta }^{\prime }$ be a basis of $W$

Then, the associated matrix to $T$ is:

$$A=[T{]}_{\beta }^{{\beta }^{\prime }}$$

…and the associated matrix to $T^{-1}$ is its inverse matrix:

$$A^{-1}=[T^{-1}{]}_{{\beta }^{\prime }}^{\beta }$$

Furthermore, let’s remember that $T\circ T^{-1}=I_W$ and that $T^{-1}\circ T=I_V$ (identity transformations). With this, then we have that:

$$[T{]}_{\beta }^{{\beta }^{\prime }}[T^{-1}{]}_{{\beta }^{\prime }}^{\beta }=[I_W{]}_{{\beta }^{\prime }}^{{\beta }^{\prime }}=I$$

$$[T^{-1}{]}_{{\beta }^{\prime }}^{\beta }[T{]}_{\beta }^{{\beta }^{\prime }}=[I_V{]}_{\beta }^{\beta }=I$$

…where $I$ denotes the respective identity matrices.