Associated Matrix of a Linear Transformation

Topics: Algebra - Linear Transformation

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Given a linear transformation $T:V\to W$, the associated matrix of $T$ is a matrix that represents $T$ on the bases of $V$ and $W$.

This matrix can be used to transform elements by having it multiply the respective coordinate vector.

Operated and composed transformations can also have an associated matrix.

Definition §

Given:

• $V,W$ vector spaces with finite dimension
• $\beta =\{x_1,\dots ,x_n\}$ a basis of $V$
• $\gamma =\{y_1,\dots ,y_n\}$ a basis of $W$
• A linear transformation $T:V\to W$

The associated matrix is a matrix $A\in M_{m\times n}(\mathbb{K})$ that contains elements $a_{ij}\in \mathbb{K}$ such that:

$$T(x_j)=\sum _{i=1}^n{a}_{ij}{y}_j,1\le j\le n$$

Such a matrix is denoted by also $[A{]}_{\beta }^{\gamma }$, and is also known as a basis change matrix (matriz de cambio de base).

Construction §

To build the associated matrix of a linear transformation $T$, we can follow three steps:

1. Transform each one of the elements in $\beta$
2. Under $\gamma$, find the coordinate vectors of the transformed elements
3. Build the matrix by using these coordinate vectors as its columns, in order.

Example

Given the following linear transformation:

$$T:P_3(\mathbb{R})\to P_2(\mathbb{R})$$

T(f(x)) := f’(x),\ f \in P_3(\mathbb{R})$$

…we have the bases of $V$ and $W$:

• $\beta =\{1,x,x^2,x^3\}$ for $P_3(\mathbb{R})$
• $\gamma =\{1,x,x^2\}$ for $P_2(\mathbb{R})$

Let’s find the associated matrix of $T$.

We can further simplify the three steps by transforming each one of the elements in $\beta$, directly representing them as a linear combination of the elements in $\gamma$:

• $T(1)=0=0(1)+0(x)+0(x^2)$
• $T(x)=1=1(1)+0(x)+0(x^2)$
• $T(x^2)=2x=0(1)+2(x)+0(x^2)$
• $T(x^3)=3x^2=0(1)+0(x)+3(x^2)$

…and then taking the coefficients to build the associated matrix:

$$[T{]}_{\beta }^{\gamma }=\left(\begin{array}{cccc}0& 1& 0& 0\\ 0& 0& 2& 0\\ 0& 0& 0& 3\end{array}\right)$$

Determinant of Associated Matrices §

(theorem)

Let $T:V\to V$. Let $\beta$ and ${\beta }^{\prime }$ be bases for $V$. of $T$ under different bases. We have that:

$$\det ([T{]}_{\beta })=\det ([T{]}_{{\beta }^{\prime }})$$