math115: do not use special colv command

properly transition from mathjax because katex doesn't support newcommand global scoping
This commit is contained in:
eggy 2022-10-12 19:10:11 -04:00
parent e08cef4a08
commit 68353b6ace

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@ -246,13 +246,12 @@ Please see [SL Math - Analysis and Approaches 2#Vectors](/g11/mcv4u7/#vectors) a
The column vector shows a vector of the form $(x, y, ...)$ from top to bottom as $(x_1, x_2, ...)$ as the number of dimensions increases.
$$
\newcommand\colv[1]{\begin{bmatrix}#1\end{bmatrix}}
\colv{x_1 \\ x_2 \\ x_3}
\begin{bmatrix}x_1 \\ x_2 \\ x_3\end{bmatrix}
$$
The zero vector is full of zeroes.
$$
\colv{0 \\ 0 \\ 0}
\begin{bmatrix}0 \\ 0 \\ 0\end{bmatrix}
$$
!!! warning
@ -394,13 +393,13 @@ In an augmented matrix, the system is consistent **if and only if** the resultan
$$\text{system is consistent}\iff\vec b = A\vec x$$
Where $\vec x$ is $\colv{x_1 \\ x_2 \\ ...}$ and $\vec a_n$ is the column vector of $A$ at $n$:
Where $\vec x$ is $\begin{bmatrix}x_1 \\ x_2 \\ ...\end{bmatrix}$ and $\vec a_n$ is the column vector of $A$ at $n$:
$$A\vec x = \vec a_1x_1 + \vec a_2x_2 + ... + \vec a_nx_n$$
**Alternatively**, the matrix-vector product can be considered a dot product such that where $\vec r_1, \vec r_2, ...$ are the rows of $A$:
$$A\vec x = \vec b = \colv{\vec r_1\bullet\vec x \\ \vec r_2\bullet\vec x \\ ... \\ \vec r_n\bullet\vec x}$$
$$A\vec x = \vec b = \begin{bmatrix}\vec r_1\bullet\vec x \\ \vec r_2\bullet\vec x \\ ... \\ \vec r_n\bullet\vec x\end{bmatrix}$$
!!! warning
- $A$ must be $m\times n$.
@ -429,7 +428,7 @@ $$A\vec x = \vec b = \colv{\vec r_1\bullet\vec x \\ \vec r_2\bullet\vec x \\ ...
$$
\begin{align*}
\vec b = \colv{-7 \\ 8} &= x_1\colv{1 \\ -1} + x_2\colv{3 \\ -4} + x_3 \colv{-2 \\ 3} \\
\vec b = \{-7 \\ 8} &= x_1\begin{bmatrix}1 \\ -1\end{bmatrix} + x_2\begin{bmatrix}3 \\ -4\end{bmatrix} + x_3 \begin{bmatrix}-2 \\ 3\end{bmatrix} \\
&= x_a\vec{a_1} + x_2\vec{a_2} + x_3\vec{a_3}
\end{align*}
$$