diff --git a/docs/ce1/math115.md b/docs/ce1/math115.md index a5521a6..266ae9e 100644 --- a/docs/ce1/math115.md +++ b/docs/ce1/math115.md @@ -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*} $$