Matrices
We introduce a shorthand notation for systems of linear equations and define basic operations for matrices.
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| Concept | Content |
|---|---|
| Systems of linear equations | On the relationship between linear combinations of vectors and systems of linear equations. |
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| Concept | Content |
|---|---|
| Linear functions | We study the properties of linear functions using the deformation of objects. |
Study Matrices #
LES: Laziness
\[\begin{aligned} a_{11}x_1 + a_{12}x_2 + \ldots + a_{1n}x_n & = b_1\\ a_{21}x_1 + a_{22}x_2 + \ldots + a_{2n}x_n & = b_2\\ & \vdots\\ a_{m1}x_1 + a_{m2}x_2 + \ldots + a_{mn}x_n & = b_m \end{aligned}\]
\[\Downarrow \text{rearrange}\]
\[\begin{aligned} \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn}\\ \end{pmatrix} \begin{pmatrix} x_1\\x_2\\ \vdots \\ x_n \end{pmatrix} = \begin{pmatrix} b_{1}\\ b_{2} \\ \vdots \\ b_{m} \end{pmatrix} \end{aligned}\]
Definition 1. Let \(m,n\in\mathbb{N}\). Then \[\mathbf{A} = \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn}\\ \end{pmatrix} ~~~ \text{ with every } a_{ij}\in\mathbb{R}\] is called a real matrix. The set of all such matrices is denoted by \(\mathbb{R}^{m\times n}\), where \(m\) is the number of rows, \(n\) is the number of columns of the matrix \(\mathbf{A}\).
Basic operations
The addition, the subtraction and the (scalar) multiplication for matrices are defined componentwisely, as for vectors.
Example 2.
\(\begin{pmatrix}1&2\\3&4\end{pmatrix} + \begin{pmatrix}5&6\\7&8\end{pmatrix} = \begin{pmatrix}1+5&2+6\\3+7&4+8\end{pmatrix} = \begin{pmatrix}6&8\\10&12\end{pmatrix}\)
\(5\cdot \begin{pmatrix}1&2\\3&4\end{pmatrix} = \begin{pmatrix}5\cdot 1&5\cdot 2\\5\cdot 3&5\cdot 4\end{pmatrix} \begin{pmatrix}5&10\\15&20\end{pmatrix}\)
Matrix-vector-product
Definition 3. Let a matrix \(\mathbf{A} = { \begin{pmatrix} \mid & & \mid \\ \mathbf{a}_1 & \ldots & \mathbf{a}_n\\ \mid & & \mid \end{pmatrix}} \in \mathbb{R}^{m\times n}\) and a vector \(\mathbf{x}=\begin{pmatrix} x_1\\ \vdots\\ x_n \end{pmatrix}\in \mathbb{R}^{n}\) be given. Then the matrix-vector-product \(\mathbf{A}\cdot\mathbf{x}\) is defined as follows: \[\mathbf{A} \cdot \mathbf{x}:= \mathbf{a}_1x_1+\ldots+\mathbf{a}_nx_n\ .\]
Example 4. \[\begin{aligned} \begin{pmatrix}1 & 0 & 0 \\0 & 2 & 1\\4 & 1 & 0\end{pmatrix}\cdot \begin{pmatrix} 2\\-1\\3 \end{pmatrix} & = \begin{pmatrix}1\\0\\4\end{pmatrix}\cdot 2 + \begin{pmatrix}0\\2\\1\end{pmatrix} \cdot (-1) + \begin{pmatrix}0\\1\\0\end{pmatrix} \cdot 3 = \begin{pmatrix} 2\\1\\7 \end{pmatrix} \end{aligned}\]
The number of columns of \(\mathbf{A}\) must equal the number of components of \(\mathbf{x}\).
The product \(\mathbf{Ax}\) is a linear combination of the columns of \(\mathbf{A}\).
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Solve the WeBWorK Exercise #
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