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There is a kernel of truth when people say that you can convert many problems in mathematics to linear algebra via representation theory

systems of linear equations

linear transformations

determinant

vector spaces

cross product

direct solution

matrix solution

particular solution

the speed of supercomputer is tested on Ax = b, pure linear algebra

matrix alphabet

linear combination = cv + dw = c\(\begin{bmatrix} 1 \\ 1 \end{bmatrix}\) + d\(\begin{bmatrix} 2 \\ 3 \end{bmatrix}\) = \(\begin{bmatrix} c + 2d \\ c + 3d \end{bmatrix}\)

column vector v is v = \(\begin{bmatrix} v_1 \\ v_2 \end{bmatrix}\) with \(v_1\) being the first component and \(v_2\) being the second

Vectors

vectors have a magnitude (length) and a direction

\(\vert\vert\) magnitude \(\vert\vert\) = \(\sqrt{x^{2}+y^{2}}\)

Or magnitude = \(\begin{pmatrix} x \\ y \end{pmatrix}\)

Basically Pythagorean’s theoream

vector addition

scalar multiplication

vector equation is b = cv + dw

Vector Algebra

Laws of algebra for the vector sum b + a = a + b commutative law a + (b + c) = (a + b) + c associative law

Laws of algebra for the scalar multiple associative law distributive law

Unit vector

Basis sets coplanar orthonormal basis standard basis set

\[\hat{a}\] \[\mu\]

Position vectors Vector geometry

centroid is the center point of any object

bisector theorem

Laws of algebra for the scalar product commutative law distributive law associative with scalar multiplication

Apollonius’s theorem

Differentiation of vector functions

Matrices

matrix multiplication

Eigenvalues and Eigenvectors

eigenvector is a special vector, also known as a characteristic vector

the amount of say, stretching, the scalar value is called eigenvalue (\(\lambda\)) the direction does not change

Av = \(\lambda\)v or (A-\(\lambda\)I)v = 0

I is identity matrix of the same size as A

we have to apply a matrix to a vector, this will transform it (rotate, stretch, shrink, reflect, etc.)

if A is a matrix, v is a vector, then Av represents the transformed version of v

if \(\lambda\) is greater than 1, the vector is stretched

if 0 < \(\lambda\) < 1, it is shrunk

Linear Algebra Problems and Solutions

Given two vectors u = \(\begin{pmatrix} 2 \\ 3 \end{pmatrix}\) and v = \(\begin{pmatrix} -1 \\ 4 \end{pmatrix}\), calculate the sum and the dot product of u and v.

The sum is: \(\begin{pmatrix} 1 \\ 7 \end{pmatrix}\)

The dot product is: 10 (from: (2×−1)+(3×4))

Given a vector w = \(\begin{pmatrix} 4 \\ -3 \end{pmatrix}\), calculate the magnitude.

The magnitude is: \(\sqrt{4^{2}+(-3)^{2}}\)