Learning Outcomes
- Calculate the change-of-basis matrix for the standard basis to a non-standard basis of \(\mathbb R^n\text{.}\)
Section 4.3 Solving Systems with Matrix Inverses (MX3)
Subsection 4.3.1 Warm Up
Activity 4.3.1.
Let \(T\colon\IR^4\to\IR^4\) be the linear bijection given by the standard matrix:
\begin{equation*}
A=\left[\begin{array}{cccc} 0 & 0 & 0 & -1 \\ 1 & 0 & -1 & -4 \\ 1 & 1 & 0 & -4 \\ 1 & -1 & -1 & 2 \end{array}\right].
\end{equation*}
(a)
If \(\vec{b}=\begin{bmatrix}1\\0\\-1\\2\end{bmatrix}\text{,}\) what is the meaning of the vector \(T^{-1}(\vec{b})\text{?}\)
(b)
Explain and demonstrate how to find the third column of \(A^{-1}\text{.}\)
Subsection 4.3.2 Class Activities
Remark 4.3.2.
So far, when working with the Euclidean vector space \(\IR^n\text{,}\) we have primarily worked with the standard basis \(\mathcal{E}=\setList{\vec{e}_1,\dots, \vec{e}_n}\text{.}\) We can explore alternative perspectives more easily if we expand our toolkit to analyze different bases.
Activity 4.3.3.
Let \(\mathcal{B}=\setList{\vec{v}_1,\vec{v}_2,\vec{v}_3}=\setList{\begin{bmatrix}1\\0\\1\end{bmatrix},\begin{bmatrix}1\\-1\\1\end{bmatrix},\begin{bmatrix}0\\1\\1\end{bmatrix}}\text{.}\)
(a)
Is \(\cal{B}\) a basis of \(\IR^3\text{?}\)
- Yes.
- No.
(b)
Since \(\cal{B}\) is a basis, we know that if \(\vec{v}\in \IR^3\text{,}\) the following vector equation will have a unique solution:
\begin{equation*}
x_1\vec{v}_1+x_2\vec{v}_2+x_3\vec{v}_3=\vec{v}
\end{equation*}
Given this, we define a map \(C_{\mathcal{B}}\colon\IR^3\to\IR^3\) via the rule that \(C_{\mathcal{B}}(\vec{v})\) is equal to the unique solution to the above vector equation. The map \(C_{\mathcal{B}}\) is a linear map.
Compute \(C_{\mathcal{B}}\left(\begin{bmatrix}1\\1\\1\end{bmatrix}\right)\text{,}\) the unique solution to
\begin{equation*}
x_1\vec{v}_1+x_2\vec{v}_2+x_3\vec{v}_3=\begin{bmatrix}1\\1\\1\end{bmatrix}.
\end{equation*}
(c)
Compute \(C_\mathcal{B}(\vec{e}_1),C_\mathcal{B}(\vec{e}_2), C_\mathcal{B}(\vec{e}_3)\) and, in doing so, write down the standard matrix \(M_\mathcal{B}\) of \(C_\mathcal{B}\text{.}\)
Definition 4.3.4.
Given a basis \(\cal{B}=\setList{\vec{v}_1,\dots, \vec{v}_n}\) of \(\IR^n\text{,}\) the change of basis/coordinate transformation from the standard basis to \(\mathcal{B}\) is the transformation \(C_\mathcal{B}\colon\IR^n\to\IR^n\) defined by the property that, for any vector \(\vec{v}\in\IR^n\text{,}\) the vector \(C_\mathcal{B}(\vec{v})\) is the unique solution to the vector equation:
\begin{equation*}
x_1\vec{v}_1+\dots+x_n\vec{v}_n=\vec{v}.
\end{equation*}
Its standard matrix is called the change-of-basis matrix from the standard basis to \(\mathcal{B}\) and is denoted by \(M_{\mathcal{B}}\text{.}\) It satisfies the following:
\begin{equation*}
M_{\mathcal{B}}=[\vec{v}_1\ \cdots\ \vec{v}_n]^{-1}.
\end{equation*}
Remark 4.3.5.
The vector \(C_\mathcal{B}(\vec{v})\) is the \(\mathcal{B}\)-coordinates of \(\vec{v}\text{.}\) If you work with standard coordinates, and I work with \(\mathcal{B}\)-coordinates, then to build the vector that you call \(\vec{v}=\begin{bmatrix}a_1\\\vdots\\a_n\end{bmatrix}=a_1\vec{e}_1+\cdots+a_n\vec{e}_n\text{,}\) I would first compute \(C_\mathcal{B}(\vec{v})=\begin{bmatrix}x_1\\\vdots\\x_n\end{bmatrix}\) and then build \(\vec{v}=x_1\vec{v}_1+\cdots+x_n\vec{v}_n\text{.}\)
In particular, notation as above, we would have:
\begin{equation*}
a_1\vec{e}_1+\cdots a_n\vec{e}_n=\vec{v}=x_1\vec{v}_1+\cdots+x_n\vec{v}_n.
\end{equation*}
Activity 4.3.6.
Let \(\vec{v}_1=\begin{bmatrix}1\\-2\\1\end{bmatrix},\ \vec{v}_2=\begin{bmatrix}-1\\0\\3\end{bmatrix},\ \vec{v}_3=\begin{bmatrix}0\\1\\-1\end{bmatrix}\text{,}\) and \(\mathcal{B}=\setList{\vec{v}_1,\vec{v}_2,\vec{v}_3}\)
(a)
Calculate \(M_{\mathcal{B}}\) using technology.
(b)
Use your result to calculate \(C_\mathcal{B}\left(\begin{bmatrix}1\\1\\1\end{bmatrix}\right)\) and express the vector \(\begin{bmatrix}1\\1\\1\end{bmatrix}\) as a linear combination of \(\vec{v}_1,\vec{v}_2,\vec{v}_3\text{.}\)
Observation 4.3.7.
Let \(T\colon\IR^n\to\IR^n\) be a linear transformation and let \(A\) denote its standard matrix. If \(\cal{B}=\setList{\vec{v}_1,\dots, \vec{v}_n}\) is some other basis, then we have:
\begin{align*}
M_\mathcal{B}AM_{\mathcal{B}}^{-1} \amp= M_\mathcal{B}A[\vec{v_1}\cdots\vec{v}_n] \\
\amp= M_\mathcal{B}[T(\vec{v}_1)\cdots T(\vec{v}_n)]\\
\amp= [C_\mathcal{B}(T(\vec{v}_1))\cdots C_\mathcal{B}(T(\vec{v}_n))]
\end{align*}
In other words, the matrix \(M_{\mathcal{B}}AM_{\mathcal{B}}^{-1}\) is the matrix whose columns consist of \(\mathcal{B}\)-coordinate vectors of the image vectors \(T(\vec{v}_i)\text{.}\) The matrix \(M_{\mathcal{B}}AM_{\mathcal{B}}^{-1}\) is called the matrix of \(T\) with respect to \(\mathcal{B}\)-coordinates.
Activity 4.3.8.
Let \(\mathcal{B}=\setList{\vec{v}_1,\vec{v}_2,\vec{v}_3}=\setList{\begin{bmatrix}1\\-2\\1\end{bmatrix},\begin{bmatrix}-1\\0\\3\end{bmatrix},\begin{bmatrix}0\\1\\-1\end{bmatrix}}\) be basis from the previous Activity. Let \(T\) denote the linear transformation whose standard matrix is given by:
\begin{equation*}
A=\begin{bmatrix}9&4&4\\6&9&2\\-18&-16&-9\end{bmatrix}.
\end{equation*}
(a)
Calculate the matrix \(M_\mathcal{B}AM_{\mathcal{B}}^{-1}\text{.}\)
(b)
The matrix \(A\) describes how \(T\) transforms the standard basis of \(\IR^3\text{.}\) The matrix \(M_\mathcal{B}AM_{\mathcal{B}}^{-1}\) describes how \(T\) transforms the basis \(\mathcal{B}\) (in \(\mathcal{B}\)-coordinates).
Which of these two descriptions of \(T\) is most helpful to you in describing/understanding/visualizing the transformation \(T\) and why?
Definition 4.3.9.
Suppose that \(A\) and \(B\) are two \(n\times n\) matrix. We say that \(A\) is similar to \(B\) if there exists an invertible matrix \(P\) that satisfies:
\begin{equation*}
PAP^{-1}=B.
\end{equation*}
The results of this section demonstrate that similar matrices can be viewed as describing the same linear transformation with respect to different bases. Specifically, if \(A\) describes a transformation with respect to the standard basis of \(\IR^n\text{,}\) then the matrix \(B\) describes the same linear transformation with respect to the basis consisting of the columns of \(P^{-1}\text{.}\)
Subsection 4.3.3 Individual Practice
Activity 4.3.10.
Suppose that \(T\colon\IR^3\to\IR^3\) is a linear transformation and you knew that \(\mathcal{B}=\{\vec{v}_1,\vec{v}_2,\vec{v}_3\}\) was a basis of \(\IR^3\) that satisfied:
\begin{equation*}
T(\vec{v}_1)=3\vec{v}_1,\ T(\vec{v}_2)=-5\vec{v}_2,\ T(\vec{v}_3)=7\vec{v}_3.
\end{equation*}
If \(A\) is the standard matrix of \(T\text{,}\) do you have enough information to determine the matrix \(M_{\mathcal{B}}AM_{\mathcal{B}}^{-1}\text{?}\) If yes, write it down; if not, describe what additional information is needed.
Subsection 4.3.4 Videos
Video coming soon to this YouTube playlist.
1
www.youtube.com/watch?v=kpOK7RhFEiQ&list=PLwXCBkIf7xBMo3zMnD7WVt39rANLlSdmjExercises 4.3.5 Exercises
Exercises available at
https://tbil.org/preview/linear-algebra/exercises/#/bank/MX3/.Subsection 4.3.6 Mathematical Writing Explorations
Activity 4.3.11.
Suppose that \(A\) is similar to \(B\text{.}\) Prove that \(B\) is also similar to \(A\text{.}\) Thus, we may simply that \(A\) and \(B\) are similar matrices.
Subsection 4.3.7 Sample Problem and Solution
Sample problem Example B.1.20.
