Lecture 12/13: Matrix Representation of Linear Transformations
Recall \(A \in M_{m \times n}\) defines a linear map
We will show that any linear map \(T: V \rightarrow W\) between finite dimensional spaces can be represented by a matrix. For example the following linear transformations
and
are examples of linear transformations with finite dimensional vector spaces that can be represented by a matrix. But in order to show this, we will start with expressing vectors uniquely relative a basis in a vector space.
Coordinate Expression for a vector
Recall that if \(\beta\) is a basis for \(V\), then for any \(v \in V\), \(v\) can be expressed uniquely as an element of \(Span(\beta)\). We proved this previously for any vector space whether infinite or finite dimensional. But if the basis is finite (\(\beta = \{v_1, ... , v_n\}\)), then for every \(v \in V\), \(v\) can be expressed uniquely in the form,
The coefficients \(a_1,...a_n\) are unique for \(v\). We want to think of this relationship as a map.
Example
Let \(V = \mathbf{R}^2, v = (2,1), \beta = \{(1,0), (0,1)\}\) and \(\beta' = \{(1,1), (1,-1)\}\).
These \(a_1\) and \(a_2\) are the coefficients that appear in writing \(v\) as a linear combinations of the vectors in the basis,
From this, we get
Of course we can use an augmented matrix and put the matrix in a reduced row echelon form to solve the system and find that the coefficients are
So we can think of \([\quad]_{\beta'}\) as a map:
The idea is that each basis \(\beta\) of \(V\) with dimension \(\dim V = n\) gives a map that identifies \(V\) with \(\mathbf{R}^n\). This map is linear, 1-1 and onto. “Bijective Correspondance”
Matrix Representation of linear transformations
Consider now \(T: V \rightarrow W\) linear and both \(V\) and \(W\) are finite dimensional. We know we can identify \(V\) with \(\mathbf{R}^n\) and we can identify \(W\) with \(\mathbf{R}^m\) but now we want to represent \(T\) as a matrix. Let \(\beta = \{v_1, ..., v_n\}\) be the basis for \(V\) and let \(\gamma = \{w_1, ..., w_n\}\) be the basis for \(W\). For \(v \in V\), we have
(This is the image of \(v\) which can be expressed uniquely in terms of the vectors of the basis \(\gamma\) because we’re in the vector space \(W\) now.) \(v\) itself is a linear combination of the basis vectors of \(\beta\). So for each \(v_j\) we have,
From this, we now have this definition,
Remark: The column vectors of \(T\) are the images of the basis vectors \(\beta\) written with respect to \(\gamma\).
Example
Let
Let \(\beta = \{1, x, x^2, x^3\}\) be the standard basis of \(P_3\). and \(\gamma = \{1, x, x^2\}\) be the standard basis of \(P_2\). We want to compute \([T_d]_{\beta}^{\gamma}\). The first thing that we want to do is to apply \(T\) on the vectors of the basis \(\beta\) so,
Next, we want to write these with respect to the basis \(\gamma = \{1,x,x^2\}\) meaning that we want to express each result as a linear combination of the vectors in the basis \(\gamma\).
So the final matrix is
(Note: We earlier said that any linear transformation between finite dimensional vector spaces can be represented by a matrix. \(L_A\) is a linear transformation. So what is the matrix representation of it? The claim is that as long as you use the standard basis for \(\mathbf{R}^n\) and \(\mathbf{R}^m\), then the representation is just \(A\) itself!)
Proof: \(\beta\) is the standard basis of \(\mathbf{R}^n\) so \(\beta = \{e_1,...,e_n\}\) where
\(e_1 = \begin{pmatrix}
1 & 0 & ... & 0
\end{pmatrix}^t\)
We can then find \([L_A]_{\beta}^{\gamma}\) by applying \(L_A\) on the vectors of \(\beta\) and then re-writing them with respect to the basis \(\gamma\),
But then consider \([L_A(e_1)]_{\gamma}\). We know \(L_A\) by definition takes a vector \(\bar{x}\) and produces \(A\bar{x}\). So \([L_A(e_1)]_{\gamma}\) = \([Ae_1]_{\gamma}\). Moreover, \(Ae_1 = \bar{a}_1e_{11}+\bar{a}_2e_{12}+...+\bar{a}_ne_{1n}\) where \(\bar{a}_1,...,\bar{a}_n\) are the columns of \(A\). But all the coefficients of \(e_1\) are zero except for the first one. So \(Ae_1 = 1\bar{a}_1\) which is just the first column of \(A\). Finally, the coordinate representatives of the first column of \(A\) with respect to basis \(\gamma\) is just the first column of \(A\) since it’s the standard basis. Similarly, we can use the same argument to find out the remaining columns and so
So far, we’ve taken a linear transformation from a vector space \(V\) to another \(W\) and represented it with a matrix \([T]_{\beta}^{\gamma}\). Above, we can also take a matrix \(A\) and turn it into a linear map \(L_A\) and if we do so with the standard bases \(\mathbf{R}^n\) and \(\mathbf{R}^m\), we can recover the matrix \(A\).
Linear Transformation as a Matrix Multiplication
For all \(v \in V\), $$ \begin{align*} [T(v)]_{\gamma} = [T]_{\beta}^{\gamma} [v]_{\beta} \end{align*} $$ \(T\) acts like matrix multiplication.
Proof: Let \(\beta=\{v_1,...,v_n\}\), \(\gamma=\{w_1,...,w_m\}\). We can write \(v\) below as a linear combination of the basis vectors in \(\beta\). We can then apply \(T\) which is linear.
But this map \([ ]_{\gamma}\) is also linear. Therefore, we can distribute it in
The expression above is exactly matrix multiplication so we can re-write this as
This is exactly what we wanted to show. \(\blacksquare\)
Example
Let’s continue the same example from earlier where we found the matrix representative for the linear transformation
Suppose we have the function \(f(x) = 3x^3 + 4x + 5\) in \(P_3\). We can express this function in terms of the basis vectors in \(\beta = \{1, x, x^2, x^3\}\) as the coefficients vector \((5, 4, 0, 3)\). So now, let’s multiply this vector by the linear transformation matrix
This gives us the coefficients vector with respect to the basis \(\gamma\). This means that the function will be \(4 + 9x^2\) which is exactly what we would get if manually applied the transformation on \(f(x)\) to get \(f'(x)\) but now instead we have a matrix to do this.
References
- Math416 by Ely Kerman