Some Linear Algebra Concepts:

Displaying a vector:
$$\vec{x} = \begin{pmatrix}8\\6\\7\\5\\3\end{pmatrix}$$

Vector addition:
$$A = [a_{1}, a_{2}, \dotsc, a_{n}]$$
$$B = [b_{1}, b_{2}, \dotsc, b_{n}]$$
$$A + B = [a_{1} + b_{1}, a_{2} + b_{2}, \dotsc, a_{n} + b_{n}]$$

Calculating vector length: The length of a vector is found by squaring each component, summing them, and taking the square root of the sum. If $\vec{v}$ is a vector, its length is $\lvert\vec{v}\rvert$.

$$\lvert\vec{v}\rvert = \sqrt{\sum_{i=1}^{n}{x_i^2}}$$

The length (magnitude) of vectors:
\begin{align*}
\|\mathbf{v}\| \text{ or } \left\|\frac{a}{b}\right \|
\end{align*}

Scalar multiplication: Multiplying a scalar (real number) to a vector means multiplying every component by that real number to yield a new vector.

$$\vec{v} = [3, 6, 8, 4] \times 1.5 = [4.5, 9, 12, 6]$$

Inner product: Multiplying a scalar (real number) to a vector means multiplying every component by that real number to yield a new vector. The inner product of two vectors (also called the dot product or scalar product) defines multiplication of vectors. The inner product of two vectors is denoted by $(\vec{v_{1}}$, $\vec{v_{2}})$ or $\vec{v_{1}} \cdot \vec{v_{2}}$.

$$(\vec{x}, \vec{y}) = \vec{x} \cdot \vec{y} = \sum_{i=1}^{n}{x_{i}y_{i}}$$
For example, if $\vec{x} = [1, 6, 7, 4]$ and $\vec{y} = [3, 2, 8, 3]$, then
$$\vec{x} \cdot \vec{y} = 1(3) + 6(2) + 7(8) + 4(3) = 83$$

Orthogonality: Two vectors are orthogonal to each other if their inner product equals zero. For example, the vectors $[2, 1, -2, 4]$ and $[3, -6, 4, 2]$ are orthogonal, because
$$[2, 1, -2, 4] \cdot [3, -6, 4, 2] = 2(3) + 1(-6) - 2(4) + 4(2) = 0$$

Normal vector: A normal vector (or unit vector) is a vector of length 1. If you divide each component by its vector length, you get the normal/unit vector.

If $\vec{v} = [2, 4, 1, 2]$, then
$${\lvert}\vec{v}{\rvert} = \sqrt{2^2 + 4^2 + 1^2 + 2^2} = \sqrt{25} = 5$$

Then $\vec{u} = [2/5, 4/5, 1/5, 2/5]$ is a normal vector, because:
$${\lvert}\vec{u}{\rvert} = \sqrt{(2/5)^2 + (4/5)^2 + (1/5)^2 + (2/5)^2} = \sqrt{25/25} = 1$$

Orthonormal vectors: Vectors of unit length that are orthogonal to each other are said to be orthonormal. For example:
$$\vec{u} = [2/5, 4/5, 1/5, 2/5]$$
and
$$\vec{v} = [3 / \sqrt{65}, -6 / \sqrt{65}, 4 / \sqrt{65}, 2 / \sqrt{65}]$$
are orthonormal, because:
$${\lvert}\vec{u}\rvert = \sqrt{(2/5)^2 + (4/5)^2 + (1/5)^2 + (2/5)^2} = 1$$
$${\lvert}\vec{v}\rvert = \sqrt{(3 / \sqrt{65})^2 + (-6 / \sqrt{65})^2 + (4 / \sqrt{65})^2 + (2 / \sqrt{65})^2} = 1$$
$$\vec{u} \cdot \vec{v} = \frac{6}{5\sqrt{65}} - \frac{6}{5\sqrt{65}} - \frac{8}{5\sqrt{65}} + \frac{8}{5\sqrt{65}} = 0$$

Square matrix: A matrix is square when $m = n$. To designate the size of a square matrix with $n$ rows and columns, it is called $n$-square. Here a 3-square matrix:

$$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix}$$

Transpose: The transpose of matrix $A$ is $A^T$:
$$A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$$
$$A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}$$

Matrix multiplication: It is possible to multiply two matrices only when the number of rows of the first matrix matches the number of columns of the second matrix.

$$AB = \begin{bmatrix} 2 & 1 & 4 \\ 1 & 5 & 2 \end{bmatrix} \begin{bmatrix} 3 & 2 \\ -1 & 4 \\ 1 & 2 \end{bmatrix} = \begin{bmatrix} 9 & 16 \\ 0 & 26 \end{bmatrix}$$

Example:

$$ab_{11} = \begin{bmatrix} 2 & 1 & 4\end{bmatrix}\begin{bmatrix} 3 \\ -1 \\ 1 \end{bmatrix} = 2(3)\ +\ 1(-1)\ +\ 4(1) = 9$$
$$ab_{12} = \begin{bmatrix} 2 & 1 & 4\end{bmatrix}\begin{bmatrix} 2 \\ 4 \\ 2 \end{bmatrix} = 2(2)\ +\ 1(4)\ +\ 4(2) = 16$$
$$ab_{21} = \begin{bmatrix} 1 & 5 & 2\end{bmatrix}\begin{bmatrix} 3 \\ -1 \\ 1 \end{bmatrix} = 1(3)\ +\ 5(-1)\ +\ 2(1) = 0$$
$$ab_{22} = \begin{bmatrix} 1 & 5 & 2\end{bmatrix}\begin{bmatrix} 2 \\ 4 \\ 2 \end{bmatrix} = 1(2)\ +\ 5(4)\ +\ 2(2) = 26$$

Identity matrix: The identity matrix $(I)$ is a square matrix where all components are $0$ expected for components on the diagonal, which are equal to $1$. If you multiple a matrix with an identity matrix, you end up with the matrix:

$$AI = \begin{bmatrix}2 & 4 & 6 \\ 1 & 3 & 5 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} = \begin{bmatrix}2 & 4 & 6 \\ 1 & 3 & 5 \end{bmatrix}$$

Orthogonal matrix: Matrix $A$ is orthogonal if $AA^T = A^TA = I$. Matrix $A$ is a symmetric matrix since it is equal to its transpose.
$$A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3/5 & -4/5 \\ 0 & 4/5 & 3/5 \end{bmatrix}$$
$A$ is orthogonal, because:
$$AA^T = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3/5 & -4/5 \\ 0 & 4/5 & 3/5 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & 3/5 & 4/5 \\ 0 & -4/5 & 3/5 \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Diagonal matrix: A diagonal matrix A is a square matrix where all the entries $a_{ij}$ are 0 when $i\neq j$.

$$A = \begin{bmatrix} a_{11} & 0 & 0 & 0 \\ 0 & a_{22} & 0 & 0 \\ 0 & 0 & a_{33} & 0 \\ 0 & 0 & 0 & a_{nn} \end{bmatrix}$$

The trace of an $n \times n$ square matrix is defined to be the sum of the elements on the main diagonal.

$$tr(A) = \sum^{n}_{i=1} a_{ii} = a_{11} + a_{22} + \dots + a_{nn}$$

Determinant: A determinant is a function of a square matrix that reduces it to a single number. The determinant of a matrix $A$ is denoted ${\lvert}A\rvert$ or $det(A)$. If $A$ consists of one element $a$, then ${\lvert}A\rvert = a$. If $A$ is a 2 x 2 matrix, then
$${\lvert}A\rvert = \left| \begin{array}{cc} a & b \\ c & d \end{array} \right| = ad - bc$$
The determinant of:
$$A = \begin{bmatrix} 4 & 1 \\ 1 & 2 \end{bmatrix}$$
is:
$${\lvert}A\rvert = \left| \begin{array}{cc} 4 & 1 \\ 1 & 2 \end{array} \right| = 4(2) - 1(1) = 7$$
When the determinant of a matrix is $0$, the inverse of the matrix does not exist.

Eigenvectors and eigenvalues: An eigenvector is a nonzero vector that satisfies the equation
$$A\vec{v} = \lambda\vec{v}$$
where $A$ is a square matrix, $\lambda$ is a scalar, and $\vec{v}$ is the eigenvector. $\lambda$ is called an eigenvalue.
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