see? eifueo is important. otherwise you forget what a rref is on the test which screws you over

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MATH 115: Linear Algebra

Set theory

!!! definition - Natural numbers (\(\mathbb N\)) are all integers greater than zero. - Integers (\(\mathbb Z\)) are all non-decimal numbers. - Rational numbers (\(\mathbb Q\)) are all numbers representable as a fraction. - Irrational numbers are all real numbers not representable as a fraction. - Real numbers (\(\mathbb R\)) are all rational or irrational numbers.

The subset sign (\(\subseteq\)) indicates that one set is strictly within another. The not subset sign (\(\not\subseteq\)) indicates that at least one element in the first set is not in the second.

!!! example - Natural numbers are a subset of integers, or \(\mathbb N \subseteq \mathbb Z\). - Integers are not a subset of natural numbers, or \(\mathbb Z \not\subseteq \mathbb N\).

!!! warning The subset sign is not to be confused with the element of sign (\(\in\)), as the former only applies to sets while the latter only applies to elements.

Sets can be subtracted with a backslash (\), returning a set with all elements in the first set not in the second.

!!! example The set of irrational numbers can be represented as the difference between the real and rational number sets, or: \[\mathbb R \backslash \mathbb Q\]

Complex numbers

A complex number can be represented in the form: \[x+yj\]

where \(x\) and \(y\) are real numbers, and \(j\) is the imaginary \(\sqrt{-1}\) (also known as \(i\) outside of engineering). This implies that every real number is also in the set of complex numbers as \(y\) can be set to zero.

!!! definition - \(Re(z)\) is the real component of complex number \(z\). - \(Im(z)\) is the imaginary component of complex number \(z\).

These numbers can be treated effectively like any other number.

Properties of complex numbers

All of these properties can be derived from expanding the standard forms.

Where \(z=x+yj\) and \(w=a+bj\):

\[ \begin{align*} zw&=(ax-by)+(bx+ay)j \\ \frac{1}{z} &= \frac{x}{x^2+y^2} - \frac{y}{x^2+y^2}j \\ z^0 &= 1 \end{align*} \]

??? example If \(z=2+5j\) and \(w=1+3j\): \[ \begin{align*} \frac{z}{w} &= (2+5j)(\frac{1}{1+9}-\frac{3}{1+9}j) \\ &= (2+5j)(\frac{1}{10}-\frac{3}{10j}) \\ &= \frac{1}{5}-\frac{3}{5}j+\frac{1}{2}j+\frac{3}{2} \\ &= \frac{17}{10}-\frac{1}{10}j \end{align*} \]

??? example To solve for \(z\) in \(z^2+4=0\): \[ \begin{align*} (x+yj)^2&=-4 \\ x^2+2xyj - y^2 &= -4 + 0j \\ (x^2-y^2) + 2xyj &= -4+0j \\ \\ ∵ x, y \in \mathbb R: 2xyj &= 0j \\ ∴ \begin{cases} x^2-y^2=-4 \\ 2xy = 0 \end{cases} \\ \\ x=0 &\text{ or } y=0 \\ \pu{if } x=0&: y =\pm 2 \\ \pu{if } y=0&: \text{no real solutions} \\ \\ ∴ x&=0, y=\pm 2 \\ z&=\pm 2j \end{align*} \]

??? example To solve for \(z\) in \(z^2=5+12j\): \[ \begin{align*} (x+yj)^2&=5+12j \\ (x^2-j^2)+2xyj = 5+12j \\ \\ \begin{cases} x^2-y^2=5 \\ 2xy = 12 \end{cases} \\ \\ y &= \frac{6}{x} \\ x^2 - \frac{6}{x}^2 &= 5\\ x^4 - 36 - 5x^2 &= 0 \\ x^2 &= 9, -4, x\in \mathbb R \\ x &= 3, -3 \\ y &= 2, -2 \\ z &= 3+2j, -3-2j \end{align*} \]

Conjugates

The conjugate of any number can be written with a bar above it. \[\overline{x+yj} = x-yj\]

The conjugate of a conjugate is the original number. \[\overline{\overline{ z}} = z\]

\(z\) is a real number if and only if its conjugate is itself. \[z\in\mathbb R \iff \overline{z}=z\]

\(z\) is purely imaginary if and only if its conjugate is the negative version of itself. \[z\in\text{only imaginary} \iff \overline{z}=-z\]

Conjugates are flexible and can almost be treated as just another factor. \[ \begin{align*} \overline{z+w}&=\overline{z}+\overline{w} \\ \overline{zw}&=(\overline{z})(\overline{w}) \\ \overline{z^k}&=\overline{z}^k \\ \overline{\biggr(\frac{z}{w}\biggr)} &= \frac{\overline{z}}{\overline{w}}, w\neq 0 \end{align*} \]

Modulus

The modulus of a number is represented by the absolute value sign. It is equal to its magnitude if the complex number were a vector. \[|z| = \sqrt{x^2+y^2}\]

!!! example The modulus of complex number \(2-j\) is: \[ \begin{align*} |2-j|&=\sqrt{2^2+(-1)^2} \\ &= -5 \end{align*} \]

If there is no imaginary component, a complex number’s modulus is its absolute value. \[z\in\mathbb R: |z|=|Re(z)|\]

Complex numbers cannot be directly compared because imaginary numbers have no inequalities, but their moduli can — the modulus of one complex number can be greater than another’s.

Properties of moduli

These can be also be manually derived.

If the modulus is zero, the complex number is zero. \[|z|=0 \iff z=0\]

The modulus of the conjugate is equal to the modulus of the original. \[|\overline{z}| = |z|\]

The number multiplied by the conjugate modulus is the square of the modulus. \[z|\overline{z}|=|z|^2\]

Moduli are also almost just a factor: \[ \begin{align*} \biggr|\frac{z}{w}\biggr| &= \frac{|z|}{|w|}, w \neq 0 \\ |zw| &= |z||w| \end{align*} \]

The moduli of the sum is always less than the sum of the moduli of the individual numbers — this is also known as the triangle inequality theorem.

\[|z+w| \leq |z|+|w|\]

Geometry

In setting the x- and y-axes to the imaginary and real components of a complex number, complex numbers can be represented almost as vectors.

(Source: Wikimedia Commons, GNU FGL 1.2 or later)

The complex number \(x+yj\) will be on the point \((x, y)\), and the modulus is the magnitude of the vector. Complex number moduli can be compared graphically if their points lie within a drawn circle centred on the origin with a point on another vector.

Polar form

The variable \(r\) is equal to the modulus of a complex number \(|z|\).

From the Pythagorean theorem, the polar form of a complex number can be expressed using the angle of the modulus to the real axis. Where \(\theta\) is the angle of the modulus to the real axis: \[z=r(\cos\theta + j\sin\theta)\]

Trigonometry can be used to calculate \(\cos\theta\) and \(\sin\theta\) as \(\cos\theta = \frac{x}{r}\) and \(\sin\theta = \frac{y}{r}\).

!!! example \(1+\sqrt{3}j=2\big(\cos\frac{\pi}{3} + j\sin\frac{\pi}{3}\big)\)

!!! warning The polar form is not unique because going around 360° results in the same vector. Where \(k\) is any integer. \[r(\cos\theta + j\sin\theta) = r(\cos(\theta+2k\pi) + j\sin(\theta+2k\pi))\]

The polar form is useful for the multiplication of complex numbers.

Because of the angle sum identities: \[z_1z_2=r_1r_2(\cos(\theta_1+\theta_2) + j\sin(\theta_1+\theta_2))\]

This can be extrapolated into Moivre’s theorem: \[z^n=r^n(\cos(n\theta) + j\sin(n\theta))\]

To determine the roots of a complex number, Moivre’s theorem can be used again: \[\sqrt[n]{z} = \sqrt[n]{r}\big(\cos\big(\frac{\theta + 2k\pi}{n}\big) + j\sin\big(\frac{\theta + 2k\pi}{n}\big)\big)\]

where \(k\) is every number in the range \([0, n-1], k\in\mathbb Z\).

!!! example To find all answers for \(w^5=-32\): \[ \begin{align*} w^5 &= 32(\cos\theta + \sin\theta) \\ w_k &= \sqrt[5]{32}\biggr[\cos\biggr(\frac{\pi + 2k\pi}{5}\biggr) + j\sin\biggr(\frac{\pi+2k\pi}{5}\biggr)\biggr] w_0 &= 2\biggr(\cos\frac{\pi}{5} + j\sin\frac{\pi}{5}\biggr) = 2e^{j\frac{\pi}{5}} \\ w_1 &= 2\biggr(\cos\frac{3\pi}{5} + j\sin\frac{3\pi}{5}\biggr) = 2e^{j\frac{3\pi}{5}} \\ w_2 &= 2(\cos\pi + j\sin\pi = 2e^{j\pi} \\ \\ \text{etc.} \end{align*} \]

The exponential form of a complex number employs Euler’s identity: \[ \begin{align*} e^{j\pi} &= -1 \\ e^{j\pi} &= \cos\theta + j\sin\theta \\ z &= re^{j\pi} \end{align*} \]

Proofs

!!! example

Vectors

Please see SL Math - Analysis and Approaches 2#Vectors and SL Physics 1#1.3 - Vectors and scalars for more information.

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.

\[ \newcommand\colv[1]{\begin{bmatrix}#1\end{bmatrix}} \colv{x_1 \\ x_2 \\ x_3} \]

The zero vector is full of zeroes. \[ \colv{0 \\ 0 \\ 0} \]

!!! warning Vectors of different dimensions cannot be compared — the missing dimensions cannot be treated as 0.

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 \(\vec y\):

\[\vec{y} = d\vec{a} + e\vec{b} + f\vec{c}\]

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.

\[||\vec x|| = \sqrt{x_1 + x_2 + x_3}\]

Properties of norms

\[ |c|\cdot ||\vec x|| = ||c\vec x|| \\ ||\vec x + \vec y|| \leq ||\vec x|| + ||\vec y|| \]

Dot product

Please see SL Math - Analysis and Approaches 2#Dot product for more information.

The Cauchy-Schwartz inequality states that the magnitude of the dot product is less than the product. \[ |\vec x\bullet\vec y|\leq||\vec x||\cdot||\vec y|| \]

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 acute.

Complex vectors

The set of complex vectors \(\mathbb C^n\) is like \(\mathbb R^n\) but for complex numbers.

The norm of a complex vector must be a real number. Therefore:

\[ \begin{align*} ||\vec z|| &= \sqrt{|z_1|^2 + |z_2|^2 + ...} \\ &= \sqrt{\overline{z_1}z_1 + \overline{z_2}z_2 + ...} \end{align*} \]

The complex inner product is the dot product between a conjugate complex vector and a complex vector.

\[ \begin{align*} \langle\vec z,\vec w\rangle &= \overline{\vec z}\bullet\vec w \\ &= \overline{z_1}w_1 + \overline{z_2}w_2 + ... \end{align*} \]

Properties of the complex inner product

\(||\vec z||^2 = \langle\vec z, \vec z\rangle\)
\(\langle\vec z, \vec w\rangle = \overline{\langle\vec w, \vec z\rangle}\)
\(\langle a\vec z, \vec w\rangle = \overline{a}\langle\vec z, \vec w\rangle\)
\(\langle\vec u + \vec z,\vec w\rangle = \langle\vec w,\vec u\rangle + \langle\vec z, \vec u\rangle\)

!!! definition - 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 \(d\).

\[n_1x_1+n_2x_2 + n_3x_3 = n_1a+n_2b+n_3c\]

Please see SL Math - Analysis and Approaches 2#Vector projections for more information.

Similarly, the component of \(\vec a\) in the direction perpendicular to \(\vec b\) is related to the vector projection.

\[ Perp_{\vec b}\vec a = \vec a - Proj_{\vec b}\vec a \\ |Perp_{\vec b}\vec a = |\vec a|\sin\theta \]

Matrices

Please see SL Math - Analysis and Approaches 2#Matrices for more information.

!!! definition - A leading entry is the first non-zero entry in a row. - A matrix is underdetermined if there are fewer variables than rows. - A matrix is overdetermined if there are more variables than rows.

Vectors can be expressed as matrices with each dimension in its own row. If there is a contradiction in the system, it is inconsistent.

The row echelon form of a matrix makes a system rapidly solvable by effectively performing elimination on the system until it is nearly completed.

!!! example The following is a vector in its row echelon form.

$$
A=
\left[\begin{array}{rrrr | r}
1 & 0 & 2 & 3 & 2 \\
0 & 0 & 1 & 3 & 4 \\
0 & 0 & 0 & -2 & -2
\end{array}\right]
$$

The reduced row echelon form of a matrix makes a system even more rapidly solvable by performing even more elimination on the system such that each leading variable is equal to one, and that variable is the only variable in the coefficient matrix.

The rank of a matrix is equal to the number of leading entries any row echelon form. \[\text{rank}(A)\]

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. \[\text{system is consistent } \iff \text{rank}(A) = \text{rank}(A|\vec b)\]

In addition, for resultant vectors with \(m\) dimensions, the system is only consistent if \(\text{rank}(A) = m\)

Each variable \(x_n\) is a leading variable if there is a leading entry in \(A\). Otherwise, it is a free variable. Systems with free variables have infinite solutions and can be represented by a vector parameter.

!!! example TODO: LEARN example

Matrix-vector product

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.

\[\text{system is consistent}\iff\vec b = A\vec x\]

Where \(\vec x\) is \(\colv{x_1 \\ x_2 \\ ...}\) 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 \(A\):

\[A\vec x = \vec b = \colv{\vec r_1\bullet\vec x \\ \vec r_2\bullet\vec x \\ ... \\ \vec r_n\bullet\vec x}\]

!!! warning - \(A\) must be \(m\times n\). - \(\vec x\) must be in \(\mathbb R^n\) (number of columns) - \(\vec b\) must be in \(\mathbb R^m\) (number of rows)

!!! example The system below:

$$
\begin{align*}
&x_1 &+ &3x_2 &- &2x_3 &= &-7 \\
-&x_1 &- &4x_2 &+ &3x_3 &= &8
\end{align*}
$$

is equivalent to the augmented matrix:
$$
\left[\begin{array}{rrr | r}
1 & 3 & -2 & -7 \\
-1 & -4 & 3 & 8
\end{array}\right]
$$

which is consistent if and only if, where $\vec{a_1}, \vec{a_2}, \vec{a_3}$ are the column vectors of $A$:

$$
\begin{align*}
\vec b = \colv{-7 \\ 8} &= x_1\colv{1 \\ -1} + x_2\colv{3 \\ -4} + x_3 \colv{-2 \\ 3} \\
&= x_a\vec{a_1} + x_2\vec{a_2} + x_3\vec{a_3}
\end{align*}
$$

The matrix-vector product is distributive, so the following properties are true.

\(A(\vec x + \vec y) = A\vec x + A\vec y\)
\((A+B)\vec x = A\vec x + B\vec x\)
\(A(c\vec x) = cA\vec x\)

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.

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 \(\vec e_i\).

\[ \begin{align*} I_4 &= \left[\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right] \\ &= [\begin{array}{} \vec e_1 & \vec e_2 & \vec e_3 & \vec e_4\end{array}] \end{align*} \]

Matrix equality

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:

\[ \text{Since }A\vec e_i = B\vec e_i \text{ for }i = 1, ... n: \\ A\vec e_i = \vec a_i, B\vec e_i = \vec b_i \\ ∴ \vec a_i = \vec b_i\text{ for } i=1, ... n,\text{ thus } A=B. \]

Flow

!!! definition - A network is a system of junctions connected by directed lines, similar to a directed graph.

In a junction, the flow in must equal the flow out. A network that follows the junction rule is at equilibrium.

In an electrical diagram, if a reference direction is selected, flow going opposite the reference direction is negative.

Matrices can be applied by applying the junction rule to systems with equal flow in and flow out for each of the smaller systems (i.e., not trying to meet every point)

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