§ 9. The Matrix Representation of a Linear Transformation

Coordinate Vector and Matrix Representation of Linear Transformation

Definition 9.1 : Ordered Basis

Let \(V\) be a finite-dimensional vector space. An ordered basis for \(V\) is a basis for \(V\) endowed with a specific order; that is, an ordered basis for \(V\) is a finite sequence of linearly independent vectors in \(V\) that generates \(V\).

Definition 9.2 : Standard Ordered Basis

For the vector space \(F^{n}\), we call \(\left\{e_{1}, e_{2}, \ldots, e_{n}\right\}\) the standard ordered basis for \(F^{n}\). Similarly, for the vector space \(\mathrm{P}_{n}(F)\), we call \(\left\{1, x, \ldots, x^{n}\right\}\) the standard ordered basis for \(\mathrm{P}_{n}(F)\).

Definition 9.3 : Coordinate Vector of \(x\) Relative to \(\beta\)

Let \(\beta=\left\{u_{1}, u_{2}, \ldots, u_{n}\right\}\) be an ordered basis for a finite-dimensional vector space \(V\). For \(x \in V\), let \(a_{1}, a_{2}, \ldots, a_{n}\) be the unique scalars such that

\[ x=\sum_{i=1}^{n} a_{i} u_{i} . \]

We define the coordinate vector of \(x\) relative to \(\beta\), denoted \([x]_{\beta}\), by

\[ [x]_{\beta}=\left(\begin{array}{c} a_{1} \\ a_{2} \\ \vdots \\ a_{n} \end{array}\right) \]

Example 9.4 : Example of coordinate vector relative to a basis

Let \(V=\mathrm{P}_{2}(\mathbb{R})\), and let \(\beta=\left\{1, x, x^{2}\right\}\) be the standard ordered basis for \(V\). If \(f(x)=4+6 x-7 x^{2}\), then

\[ [f]_{\beta}=\left(\begin{array}{r} 4 \\ 6 \\ -7 \end{array}\right) \]

Definition 9.5 : Matrix Representation of \(T\) in the Ordered Bases \(\beta\) and \(\gamma\)

Suppose that \(V\) and \(W\) are finite-dimensional vector spaces with ordered bases \(\beta=\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}\) and \(\gamma=\left\{w_{1}, w_{2}, \ldots, w_{m}\right\}\), respectively. Let \(T : V \rightarrow W\) be linear. Then for each \(j, 1 \leq j \leq n\), there exist unique scalars \(a_{i j} \in F, 1 \leq i \leq m\), such that

\[ T\left(v_{j}\right)=\sum_{i=1}^{m} a_{i j} w_{i} \quad \text { for } 1 \leq j \leq n \]

We call the \(m \times n\) matrix \(A\) defined by \(A_{i j}=a_{i j}\) the matrix representation of T in the ordered bases \(\beta\) and \(\gamma\) and write \(A=[T]_{\beta}^{\gamma}\). If \(V=W\) and \(\beta=\gamma\), then we write \(A=[T]_{\beta}\).

Notice that the \(j\) th column of \(A\) is simply \(\left[T\left(v_{j}\right)\right]_{\gamma}\). Also observe that if \(U: V \rightarrow W\) is a linear transformation such that \([U]_{\beta}^{\gamma}=[T]_{\beta}^{\gamma}\), then \(U=T\) by Corollary 8.18.

Example 9.6 : Example of matrix representation of a linear transformation

Let \(T: \mathrm{P}_{3}(\mathbb{R}) \rightarrow \mathrm{P}_{2}(\mathbb{R})\) be the linear transformation defined by \(T(f(x))=f^{\prime}(x)\). Let \(\beta\) and \(\gamma\) be the standard ordered bases for \(\mathrm{P}_{3}(\mathbb{R})\) and \(\mathrm{P}_{2}(\mathbb{R})\), respectively. Then

\[ \begin{aligned} T(1) & =0 \cdot 1+0 \cdot x+0 \cdot x^{2} \\ T(x) & =1 \cdot 1+0 \cdot x+0 \cdot x^{2} \\ T\left(x^{2}\right) & =0 \cdot 1+2 \cdot x+0 \cdot x^{2} \\ T\left(x^{3}\right) & =0 \cdot 1+0 \cdot x+3 \cdot x^{2} . \end{aligned} \]

So

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

Note that when \(T\left(x^{j}\right)\) is written as a linear combination of the vectors of \(\gamma\), its coefficients give the entries of column \(j+1\) of \([T]_{\beta}^{\gamma}\).

Vector Space of Linear Transformations

Definition 9.7 : Addition and Scalar Multiplication of Linear Transformations

Let \(T, U: V \rightarrow W\) be arbitrary functions, where \(V\) and \(W\) are vector spaces over \(F\), and let \(a \in F\). We define \(T+U: V \rightarrow W\) by \((T+U)(x)=T(x)+U(x)\) for all \(x \in V\), and \(a T: V \rightarrow W\) by \((a T)(x)=a T(x)\) for all \(x \in V\).

Theorem 9.8 : Collection of linear transformations form a vector space.

Let \(V\) and \(W\) be vector spaces over a field \(F\), and let \(T, U: V \rightarrow W\) be linear.

• (a) For all \(a \in F, a T+U\) is linear.
• (b) Using the operations of addition and scalar multiplication in the preceding definition, the collection of all linear transformations from \(V\) to \(W\) is a vector space over \(F\).

Proof

• (a)
  Let \(x, y \in V\) and \(c \in F\). Then

  \[ \begin{aligned} (a T+U)(c x+y) & =a T(c x+y)+U(c x+y) \\ & =a[T(c x+y)]+U(c x+y) \\ & =a[c T(x)+T(y)]+c U(x)+U(y) \\ & =a c T(x)+c U(x)+a T(y)+U(y) \\ & =c(a T+U)(x)+(a T+U)(y) . \end{aligned} \]

  So \(a T+U\) is linear.

• (b)
  Noting that \(T_{0}\), the zero transformation, plays the role of the zero vector, it is easy to verify that the axioms of a vector space are satisfied, and hence that the collection of all linear transformations from \(V\) into \(W\) is a vector space over \(F\).

Definition 9.9 : \(\mathcal{L}(V, W)\), the Vector Space of all Linear Transformations from \(V\) into \(W\)

Let \(V\) and \(W\) be vector spaces over \(F\). We denote the vector space of all linear transformations from \(V\) into \(W\) by \(\mathcal{L}(V, W)\). In the case that \(V=W\), we write \(\mathcal{L}(V)\) instead of \(\mathcal{L}(V, W)\).

Theorem 9.10 : Matrix representations respect addition and scalar multiplication.

Let \(V\) and \(W\) be finite-dimensional vector spaces with ordered bases \(\beta\) and \(\gamma\), respectively, and let \(T, U: V \rightarrow W\) be linear transformations. Then

• (a) \([T+U]_{\beta}^{\gamma}=[T]_{\beta}^{\gamma}+[U]_{\beta}^{\gamma}\) and
• (b) \([a T]_{\beta}^{\gamma}=a[T]_{\beta}^{\gamma}\) for all scalars \(a\).

Proof

Let \(\beta=\left\{v_{1}, v_{2}, \ldots, v_{n}\right\}\) and \(\gamma=\left\{w_{1}, w_{2}, \ldots, w_{m}\right\}\). There exist unique scalars \(a_{i j}\) and \(b_{i j}(1 \leq i \leq m, 1 \leq j \leq n)\) such that

\[ T\left(v_{j}\right)=\sum_{i=1}^{m} a_{i j} w_{i} \quad \text { and } \quad U\left(v_{j}\right)=\sum_{i=1}^{m} b_{i j} w_{i} \quad \text { for } 1 \leq j \leq n . \]

Hence

\[ (T+U)\left(v_{j}\right)=\sum_{i=1}^{m}\left(a_{i j}+b_{i j}\right) w_{i} \]

Thus

\[ \left([T+U]_{\beta}^{\gamma}\right)_{i j}=a_{i j}+b_{i j}=\left([T]_{\beta}^{\gamma}+[U]_{\beta}^{\gamma}\right)_{i j} . \]

So (a) is proved, and the proof of (b) is similar.

Exercise

Exercise 9.8

Let \(V\) be an \(n\)-dimensional vector space with an ordered basis \(\beta\). Define \(T: V \rightarrow F^{n}\) by \(T(x)=[x]_{\beta}\). Prove that \(T\) is linear.

Exercise 9.11

Let \(V\) be an \(n\)-dimensional vector space, and let \(T: V \rightarrow V\) be a linear transformation. Suppose that \(W\) is a \(T\)-invariant subspace of \(V\) having dimension \(k\). Show that there is a basis \(\beta\) for \(V\) such that \([T]_{\beta}\) has the form

\[ \left(\begin{array}{ll} A & B \\ O & C \end{array}\right) \]

where \(A\) is a \(k \times k\) matrix and \(O\) is the \((n-k) \times k\) zero matrix.

Exercise 9.13

Let \(V\) and \(W\) be vector spaces, and let \(T\) and \(U\) be nonzero linear transformations from \(V\) into \(W\). If \(R(T) \cap R(U)=\{0\}\), prove that \(\{T, U\}\) is a linearly independent subset of \(\mathcal{L}(V, W)\).