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math115: properly close vectors

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yet another mathjax/katex incompatibility - there will be more
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@ -259,19 +259,19 @@ $$
The standard form of a vector is written as the difference between two points: $\vec{OA}$ where $O$ is the origin and $A$ is any point. $\vec{AB}$ is the vector as a difference between two points. The standard form of a vector is written as the difference between two points: $\vec{OA}$ where $O$ is the origin and $A$ is any point. $\vec{AB}$ is the vector as a difference between two points.
If a vector can be expressed as the sum of a scalar multiple of other vectors, that vector is the **linear combination** of those vectors. Formally, $\vec{y}$ is a linear combination of $\vec{a}, \vec{b}, \vec{c}$ if and only if any **real** constant(s) multiplied by each vector return $\vec y$: If a vector can be expressed as the sum of a scalar multiple of other vectors, that vector is the **linear combination** of those vectors. Formally, $\vec{y}$ is a linear combination of $\vec{a}, \vec{b}, \vec{c}$ if and only if any **real** constant(s) multiplied by each vector return $\vec{y}$:
$$\vec{y} = d\vec{a} + e\vec{b} + f\vec{c}$$ $$\vec{y} = d\vec{a} + e\vec{b} + f\vec{c}$$
The **norm** of a vector is its magnitude or distance from the origin, represented by double absolute values. In $\mathbb R^2$ and $\mathbb R^3$, the Pythagorean theorem can be used. The **norm** of a vector is its magnitude or distance from the origin, represented by double absolute values. In $\mathbb R^2$ and $\mathbb R^3$, the Pythagorean theorem can be used.
$$||\vec x|| = \sqrt{x_1 + x_2 + x_3}$$ $$||\vec{x}|| = \sqrt{x_1 + x_2 + x_3}$$
### Properties of norms ### Properties of norms
$$ $$
|c|\cdot ||\vec x|| = ||c\vec x|| \\ |c|\cdot ||\vec{x}|| = ||c\vec{x}|| \\
||\vec x + \vec y|| \leq ||\vec x|| + ||\vec y|| ||\vec{x} + \vec{y}|| \leq ||\vec{x}|| + ||\vec{y}||
$$ $$
### Dot product ### Dot product
@ -280,13 +280,13 @@ Please see [SL Math - Analysis and Approaches 2#Dot product](/g11/mcv4u7/#dot-pr
The Cauchy-Schwartz inequality states that the magnitude of the dot product is less than the product. The Cauchy-Schwartz inequality states that the magnitude of the dot product is less than the product.
$$ $$
|\vec x\bullet\vec y|\leq||\vec x||\cdot||\vec y|| |\vec{x}\bullet\vec{y}|\leq||\vec{x}||\cdot||\vec{y}||
$$ $$
The dot product can be used to guesstimate the angle between two vectors. The dot product can be used to guesstimate the angle between two vectors.
- If $\vec x\bullet\vec y < 0$, the angle is obtuse. - If $\vec{x}\bullet\vec{y} < 0$, the angle is obtuse.
- If $\vec x\bullet\vec y > 0$, the angle is acute. - If $\vec{x}\bullet\vec{y} > 0$, the angle is acute.
### Complex vectors ### Complex vectors
@ -296,7 +296,7 @@ The **norm** of a complex vector must be a real number. Therefore:
$$ $$
\begin{align*} \begin{align*}
||\vec z|| &= \sqrt{|z_1|^2 + |z_2|^2 + ...} \\ ||\vec{z}|| &= \sqrt{|z_1|^2 + |z_2|^2 + ...} \\
&= \sqrt{\overline{z_1}z_1 + \overline{z_2}z_2 + ...} &= \sqrt{\overline{z_1}z_1 + \overline{z_2}z_2 + ...}
\end{align*} \end{align*}
$$ $$
@ -305,17 +305,17 @@ The **complex inner product** is the dot product between a conjugate complex vec
$$ $$
\begin{align*} \begin{align*}
\langle\vec z,\vec w\rangle &= \overline{\vec z}\bullet\vec w \\ \langle\vec{z},\vec{w}\rangle &= \overline{\vec{z}}\bullet\vec{w} \\
&= \overline{z_1}w_1 + \overline{z_2}w_2 + ... &= \overline{z_1}w_1 + \overline{z_2}w_2 + ...
\end{align*} \end{align*}
$$ $$
#### Properties of the complex inner product #### Properties of the complex inner product
- $||\vec z||^2 = \langle\vec z, \vec z\rangle$ - $||\vec{z}||^2 = \langle\vec{z}, \vec{z}\rangle$
- $\langle\vec z, \vec w\rangle = \overline{\langle\vec w, \vec z\rangle}$ - $\langle\vec{z}, \vec{w}\rangle = \overline{\langle\vec{w}, \vec{z}\rangle}$
- $\langle a\vec z, \vec w\rangle = \overline{a}\langle\vec z, \vec w\rangle$ - $\langle a\vec{z}, \vec{w}\rangle = \overline{a}\langle\vec{z}, \vec{w}\rangle$
- $\langle\vec u + \vec z,\vec w\rangle = \langle\vec w,\vec u\rangle + \langle\vec z, \vec u\rangle$ - $\langle\vec{u} + \vec{z},\vec{w}\rangle = \langle\vec{w},\vec{u}\rangle + \langle\vec{z}, \vec{u}\rangle$
### Cross product ### Cross product
@ -332,17 +332,17 @@ Please see [SL Math - Analysis and Approaches 2#Vector planes](/g11/mcv4u7/#vect
!!! definition !!! definition
- A **hyperplane** is an $\mathbb R^{n-1}$ plane in an $\mathbb R^n$ space. - A **hyperplane** is an $\mathbb R^{n-1}$ plane in an $\mathbb R^n$ space.
The **scalar equation** of a vector shows the normal vector $\vec n$ and a point on the plane $P(a,b,c)$ which can be condensed into the constant $d$. The **scalar equation** of a vector shows the normal vector $\vec{n}$ and a point on the plane $P(a,b,c)$ which can be condensed into the constant $d$.
$$n_1x_1+n_2x_2 + n_3x_3 = n_1a+n_2b+n_3c$$ $$n_1x_1+n_2x_2 + n_3x_3 = n_1a+n_2b+n_3c$$
Please see [SL Math - Analysis and Approaches 2#Vector projections](/g11/mcv4u7/#vector-projections) for more information. Please see [SL Math - Analysis and Approaches 2#Vector projections](/g11/mcv4u7/#vector-projections) for more information.
Similarly, the component of $\vec a$ in the direction **perpendicular to** $\vec b$ is related to the vector projection. Similarly, the component of $\vec{a}$ in the direction **perpendicular to** $\vec{b}$ is related to the vector projection.
$$ $$
Perp_{\vec b}\vec a = \vec a - Proj_{\vec b}\vec a \\ Perp_{\vec{b}}\vec{a} = \vec{a} - Proj_{\vec{b}}\vec{a} \\
|Perp_{\vec b}\vec a = |\vec a|\sin\theta |Perp_{\vec{b}}\vec{a} = |\vec{a}|\sin\theta
$$ $$
## Matrices ## Matrices
@ -375,10 +375,10 @@ The **reduced row echelon form** of a matrix makes a system even more rapidly so
The **rank** of a matrix is equal to the number of leading entries any row echelon form. The **rank** of a matrix is equal to the number of leading entries any row echelon form.
$$\text{rank}(A)$$ $$\text{rank}(A)$$
In general, $A$ represents just the coefficient matrix, while $A|\vec b$ represents the augmented matrix. In general, $A$ represents just the coefficient matrix, while $A|\vec{b}$ represents the augmented matrix.
According to the **system-rank theorem**, a system is consistent **if and only if** the ranks of the coefficient and augmented matrices are equal. According to the **system-rank theorem**, a system is consistent **if and only if** the ranks of the coefficient and augmented matrices are equal.
$$\text{system is consistent } \iff \text{rank}(A) = \text{rank}(A|\vec b)$$ $$\text{system is consistent } \iff \text{rank}(A) = \text{rank}(A|\vec{b})$$
In addition, for resultant vectors with $m$ dimensions, the system is only consistent if $\text{rank}(A) = m$ In addition, for resultant vectors with $m$ dimensions, the system is only consistent if $\text{rank}(A) = m$
@ -391,20 +391,20 @@ Each variable $x_n$ is a **leading variable** if there is a leading entry in $A$
In an augmented matrix, the system is consistent **if and only if** the resultant vector is a linear combination of the columns of the coefficient matrix. In an augmented matrix, the system is consistent **if and only if** the resultant vector is a linear combination of the columns of the coefficient matrix.
$$\text{system is consistent}\iff\vec b = A\vec x$$ $$\text{system is consistent}\iff\vec{b} = A\vec{x}$$
Where $\vec x$ is $\begin{bmatrix}x_1 \\ x_2 \\ ...\end{bmatrix}$ and $\vec a_n$ is the column vector of $A$ at $n$: Where $\vec{x}$ is $\begin{bmatrix}x_1 \\ x_2 \\ ...\end{bmatrix}$ and $\vec{a}_n$ is the column vector of $A$ at $n$:
$$A\vec x = \vec a_1x_1 + \vec a_2x_2 + ... + \vec a_nx_n$$ $$A\vec{x} = \vec{a}_1x_1 + \vec{a}_2x_2 + ... + \vec{a}_nx_n$$
**Alternatively**, the matrix-vector product can be considered a dot product such that where $\vec r_1, \vec r_2, ...$ are the rows of $A$: **Alternatively**, the matrix-vector product can be considered a dot product such that where $\vec{r}_1, \vec{r}_2, ...$ are the rows of $A$:
$$A\vec x = \vec b = \begin{bmatrix}\vec r_1\bullet\vec x \\ \vec r_2\bullet\vec x \\ ... \\ \vec r_n\bullet\vec x\end{bmatrix}$$ $$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}$$
!!! warning !!! warning
- $A$ must be $m\times n$. - $A$ must be $m\times n$.
- $\vec x$ must be in $\mathbb R^n$ (number of columns) - $\vec{x}$ must be in $\mathbb R^n$ (number of columns)
- $\vec b$ must be in $\mathbb R^m$ (number of rows) - $\vec{b}$ must be in $\mathbb R^m$ (number of rows)
!!! example !!! example
The system below: The system below:
@ -428,22 +428,22 @@ $$A\vec x = \vec b = \begin{bmatrix}\vec r_1\bullet\vec x \\ \vec r_2\bullet\vec
$$ $$
\begin{align*} \begin{align*}
\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} \\ \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} &= x_a\vec{a_1} + x_2\vec{a_2} + x_3\vec{a_3}
\end{align*} \end{align*}
$$ $$
The matrix-vector product is distributive, so the following properties are true. The matrix-vector product is distributive, so the following properties are true.
- $A(\vec x + \vec y) = A\vec x + A\vec y$ - $A(\vec{x} + \vec{y}) = A\vec{x} + A\vec{y}$
- $(A+B)\vec x = A\vec x + B\vec x$ - $(A+B)\vec{x} = A\vec{x} + B\vec{x}$
- $A(c\vec x) = cA\vec x$ - $A(c\vec{x}) = cA\vec{x}$
### Identity matrices ### Identity matrices
In a **homogeneous system** ($\vec b = \vec 0$), any linear combinations of the solutions to the system ($\vec x_1, ... \vec x_n$) are also solutions to the system. In a **homogeneous system** ($\vec{b} = \vec{0}$), any linear combinations of the solutions to the system ($\vec{x}_1, ... \vec{x}_n$) are also solutions to the system.
The identity matrix ($I_n$) is a **square matrix** of size $n$ with the value 1 along the main diagonal and 0 everywhere else. The $i$th column is equal to the $i$th row, which is known as $\vec e_i$. The identity matrix ($I_n$) is a **square matrix** of size $n$ with the value 1 along the main diagonal and 0 everywhere else. The $i$th column is equal to the $i$th row, which is known as $\vec{e}_i$.
$$ $$
\begin{align*} \begin{align*}
@ -453,20 +453,20 @@ I_4 &= \left[\begin{array}{rrrr}
0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 0 & 0 & 0 & 1
\end{array}\right] \\ \end{array}\right] \\
&= [\begin{array}{} \vec e_1 & \vec e_2 & \vec e_3 & \vec e_4\end{array}] &= [\begin{array}{} \vec{e}_1 & \vec{e}_2 & \vec{e}_3 & \vec{e}_4\end{array}]
\end{align*} \end{align*}
$$ $$
## Matrix equality ## Matrix equality
Matrices are only equal if *every* possible linear combination is equal ($A\vec x = B\vec x$ **does not mean** $A = B$). Matrices are only equal if *every* possible linear combination is equal ($A\vec{x} = B\vec{x}$ **does not mean** $A = B$).
If $A\vec x = B\vec x$ for every $\vec x\in \mathbb R^n$, then $A = B$. This can be proven using the identity matrix: If $A\vec{x} = B\vec{x}$ for every $\vec{x}\in \mathbb R^n$, then $A = B$. This can be proven using the identity matrix:
$$ $$
\text{Since }A\vec e_i = B\vec e_i \text{ for }i = 1, ... n: \\ \text{Since }A\vec{e}_i = B\vec{e}_i \text{ for }i = 1, ... n: \\
A\vec e_i = \vec a_i, B\vec e_i = \vec b_i \\ A\vec{e}_i = \vec{a}_i, B\vec{e}_i = \vec{b}_i \\
∴ \vec a_i = \vec b_i\text{ for } i=1, ... n,\text{ thus } A=B. ∴ \vec{a}_i = \vec{b}_i\text{ for } i=1, ... n,\text{ thus } A=B.
$$ $$
## Flow ## Flow