Objectives: Topics
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Matrix notation for systems of equations.
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The matrix product \(A \vec{x}\text{.}\)
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Section 1.4 The Matrix Equation \(A \vec{x} = \vec{b}\)
Handout 1.4 The Matrix Equation \(A \vec{x} = \vec{b}\)
In this section we introduce another way of expressing a linear system that we will use throughout this course.
Objectives: Goals
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Compute matrix–vector products.
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Express linear systems as vector equations and matrix equations.
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Characterize linear systems and sets of vectors using span, linear combinations, and pivots.
Below is some math notation that will be used frequently throughout the course.
| symbol | meaning |
| \(\in\) | belongs to |
| \(\mathbb{R}^{n}\) | the set of vectors with \(n\) real-valued elements |
| \(\mathbb{R}^{m \times n}\) | the set of real-valued matrices with \(m\) rows and \(n\) columns |
Example: the notation \(\vec{x} \in \mathbb{R}^{5}\) means that \(\vec{x}\) is a vector with five real-valued elements.
Definition 1.33. Matrix Vector Product as a Linear Combination.
Let \(A\) be an \(m \times n\) matrix with columns \(\vec{a}_1,\dotsc,\vec{a}_n\) and let \(\vec{x} \in \mathbb{R}^{n}\text{.}\) The matrix vector product \(A\vec{x}\) is the following linear combination of the columns of \(A\text{:}\)
\begin{equation*}
A \vec{x}
=
\begin{bmatrix}
\vert \ \amp \ \vert \ \amp \ \cdots \ \amp \ \vert \\
\vec{a}_1 \ \amp \ \vec{a}_2 \ \amp \ \cdots \ \amp \ \vec{a}_n \\
\vert \ \amp \ \vert \ \amp \ \cdots \ \amp \ \vert
\end{bmatrix}
\begin{bmatrix}
x_1 \\ x_2 \\ \vdots \\ x_n
\end{bmatrix}
=
x_1 \vec{a}_1 + x_2 \vec{a}_2 + \cdots + x_n \vec{a}_n.
\end{equation*}
Example 1.34.
Write the product below as a linear combination of column vectors.
\(\begin{bmatrix}
1 \ \amp \ 0 \ \amp \ -1 \\
0 \ \amp \ -3 \ \amp \ 3
\end{bmatrix}
\begin{bmatrix}
4 \\ 3 \\ 7
\end{bmatrix}
\;=\; \)
Example 1.35.
Simplify the product below.
\(\begin{bmatrix}
1 \ \amp \ 0 \ \amp \ 2 \ \amp \ 0 \ \amp \ 3 \\
0 \ \amp \ 1 \ \amp \ 0 \ \amp \ 2 \ \amp \ 0
\end{bmatrix}
\begin{bmatrix}
x_1 \\ x_2 \\ x_3 \\ x_4 \\ x_5
\end{bmatrix}
\;=\;\)
Theorem 1.36. Equivalent Formulations of a Linear System.
Let \(A\) be an \(m \times n\) matrix with columns \(\vec{a}_1,\dotsc,\vec{a}_n\text{,}\) let \(\vec{x} \in \mathbb{R}^{n}\text{,}\) and let \(\vec{b} \in \mathbb{R}^{m}\text{.}\) The solutions to
\begin{equation*}
A \vec{x} = \vec{b}
\end{equation*}
are the same as the solutions to the vector equation
\begin{equation*}
x_1 \vec{a}_1 + \cdots + x_n \vec{a}_n \;=\; \vec{b},
\end{equation*}
which are the same as the solutions to the linear system with augmented matrix
\begin{equation*}
\begin{bmatrix}
\vec{a}_1 \ \amp \ \vec{a}_2 \ \amp \ \cdots \ \amp \ \vec{a}_n \ \amp \vert \amp \ \vec{b}
\end{bmatrix}.
\end{equation*}
Theorem 1.37. The Existence of Solutions for the Matrix Equation.
The equation \(A \vec{x} = \vec{b}\) has a solution if and only if \(\vec{b}\) is a linear combination of the columns of \(A\text{.}\)
Theorem 1.38. Consistency for all \(\vec{b}\) and pivots.
The equation \(A \vec{x} = \vec{b}\) is consistent for every \(\vec{b}\) if and only if every row of \(A\) has a pivot.
Example 1.39.
For what vectors \(\vec{b} = \begin{bmatrix} b_1 \\[2pt] b_2 \\[2pt] b_3 \end{bmatrix}\) does the equation have a solution?
\begin{equation*}
\begin{bmatrix}
1 \ \amp \ 3 \ \amp \ 4 \\
2 \ \amp \ 8 \ \amp \ 4 \\
0 \ \amp \ 1 \ \amp \ -2
\end{bmatrix}
\vec{x} \;=\; \vec{b}
\end{equation*}
Example 1.40.
For what values of \(h\) is the system \(A\vec{x} = \vec{b}\) consistent for every possible choice of vector \(\vec{b}\text{?}\)
\begin{equation*}
A \;=\; \begin{bmatrix}
1 \ \amp \ h \ \amp \ 2 \\
0 \ \amp \ 6 \ \amp \ 2 \\
-3 \ \amp \ 0 \ \amp \ -4
\end{bmatrix}
\end{equation*}
We now have four equivalent ways of expressing linear systems.
A system of equations:
\begin{equation*}
\begin{aligned}
2x_1 + 3x_2 \ \amp=\ 7 \\
x_1 - x_2 \ \amp=\ 5
\end{aligned}
\end{equation*}
An augmented matrix:
\begin{equation*}
\left[
\begin{array}{rr|r}
2 \ \amp \ 3 \ \amp \ 7 \\
1 \ \amp \ -1 \ \amp \ 5
\end{array}
\right]
\end{equation*}
A vector equation:
\begin{equation*}
x_1 \begin{bmatrix} 2 \\[2pt] 1 \end{bmatrix}
\;+\;
x_2 \begin{bmatrix} 3 \\[2pt] -1 \end{bmatrix}
\;=\;
\begin{bmatrix} 7 \\[2pt] 5 \end{bmatrix}
\end{equation*}
A matrix equation:
\begin{equation*}
\begin{bmatrix}
2 \ \amp \ 3 \\
1 \ \amp \ -1
\end{bmatrix}
\begin{bmatrix}
x_1 \\ x_2
\end{bmatrix}
\;=\;
\begin{bmatrix}
7 \\ 5
\end{bmatrix}
\end{equation*}
Each representation gives us a different way to think about linear systems.
