3.2 Matrix Representation of Linear Transformations

Matrix representation of linear transformations is a powerful tool in linear algebra. It allows us to convert abstract transformations into concrete matrices, making calculations easier. This connection between transformations and matrices is key to understanding how linear algebra works in practice.

By representing transformations as matrices, we can use matrix operations to study and manipulate them. This approach lets us solve complex problems in linear algebra, from finding eigenvalues to analyzing systems of equations, all through the lens of matrices.

Matrices for Linear Transformations

Definition and Representation

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• A linear transformation $T$ from a vector space $V$ to a vector space $W$ preserves vector addition and scalar multiplication
• Every linear transformation can be represented by a matrix with respect to a choice of bases for the domain and codomain
• The matrix representation of a linear transformation $T: V \to W$ with respect to bases $B$ for $V$ and $C$ for $W$ is the matrix $A$ such that for any vector $v$ in $V$, $T(v) = A[v]_B$, where $[v]_B$ denotes the coordinate vector of $v$ with respect to the basis $B$

Matrix Dimensions and Entries

• The dimensions of the matrix $A$ are determined by the dimensions of the codomain (number of rows) and the domain (number of columns) of the linear transformation
• The entries of the matrix $A$ are determined by the images of the basis vectors of $V$ under the linear transformation $T$
  • For example, if $T: \mathbb{R}^2 \to \mathbb{R}^3$ and the standard basis vectors $e_1 = (1, 0)$ and $e_2 = (0, 1)$ are mapped to $T(e_1) = (1, 2, 3)$ and $T(e_2) = (4, 5, 6)$, then the matrix representation of $T$ with respect to the standard bases is: $A = \begin{pmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{pmatrix}$

Matrix Representations of Transformations

Finding the Matrix Representation

• To find the matrix representation of a linear transformation $T: V \to W$ with respect to bases $B$ for $V$ and $C$ for $W$, first determine the images of each basis vector in $B$ under the transformation $T$
• Express each image $T(v_i)$ as a linear combination of the basis vectors in $C$, where $v_i$ is the $i$-th basis vector in $B$
  • For instance, if $T(v_1) = 2w_1 - w_2$ and $T(v_2) = w_1 + 3w_2$, where $w_1$ and $w_2$ are basis vectors in $C$, then the columns of the matrix representation are $\begin{pmatrix} 2 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 3 \end{pmatrix}$
• The coefficients of these linear combinations form the columns of the matrix $A$, with the $i$-th column corresponding to the image of the $i$-th basis vector in $B$

Resulting Matrix

• The resulting matrix $A$ is the matrix representation of the linear transformation $T$ with respect to the chosen bases $B$ and $C$
  • In the previous example, the matrix representation of $T$ with respect to bases $B$ and $C$ is: $A = \begin{pmatrix} 2 & 1 \\ -1 & 3 \end{pmatrix}$

Transformations from Matrices

Uniqueness of Linear Transformation

• Given a matrix $A$ and bases $B$ for a vector space $V$ and $C$ for a vector space $W$, there exists a unique linear transformation $T: V \to W$ such that $A$ is the matrix representation of $T$ with respect to the bases $B$ and $C$

Finding the Linear Transformation

• To find the linear transformation $T$ corresponding to a matrix $A$, let $v$ be an arbitrary vector in $V$ and express it as a linear combination of the basis vectors in $B$
• Multiply the matrix $A$ by the coordinate vector $[v]_B$ to obtain the coordinate vector of the image $T(v)$ with respect to the basis $C$
  • For example, if $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $v = (x, y)$ in the standard basis of $\mathbb{R}^2$, then $[v]_B = \begin{pmatrix} x \\ y \end{pmatrix}$ and: $T(v) = A[v]_B = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x + 2y \\ 3x + 4y \end{pmatrix}$
• The linear transformation $T$ is defined by mapping each vector $v$ in $V$ to its image $T(v)$ obtained through matrix multiplication

Matrix Multiplication vs Composition

Composition of Linear Transformations

• Given linear transformations $S: U \to V$ and $T: V \to W$, the composition of these transformations, denoted by $T \circ S$, is a linear transformation from $U$ to $W$ defined by $(T \circ S)(u) = T(S(u))$ for all $u$ in $U$

Matrix Representations of Compositions

• Let $A$ be the matrix representation of $S$ with respect to bases $B$ for $U$ and $C$ for $V$, and let $B$ be the matrix representation of $T$ with respect to bases $C$ for $V$ and $D$ for $W$
• To prove that the matrix representation of $T \circ S$ with respect to bases $B$ and $D$ is the product $BA$, consider an arbitrary vector $u$ in $U$ and its coordinate vector $[u]_B$
• Show that $(T \circ S)(u) = T(S(u)) = B(A[u]_B) = (BA)[u]_B$ by using the definitions of matrix representations and matrix multiplication
  • For instance, if $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix}$, then: $(BA)[u]_B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} [u]_B = \begin{pmatrix} 19 & 22 \\ 43 & 50 \end{pmatrix} [u]_B$

Conclusion

• Conclude that the matrix representation of the composition $T \circ S$ with respect to bases $B$ and $D$ is the product of the matrix representations of $T$ and $S$, in that order
  • This relationship between matrix multiplication and composition of linear transformations allows for the study of linear transformations using matrix algebra