Linear Algebra

Tue 29 May 2018

Vectors

Quantity comprised of a direction and magnitude.

Span of Vectors

All the vectors that can be created from a set of vectors.

Vectors are linearly dependent if you can remove one and not reduce the span. One of the vectors can be produced by a linear combination of the others. Otherwise they are linearly independent if each vector adds a new dimension to the span.

Dot Product

aka. Inner Product of Two vectors. Used to find the angle between two vectors.

$$ \vec{x} * \vec{y} = ||\vec{x}|| * ||\vec{y}|| * cos(\theta) $$

Example:

$$ \begin{bmatrix}1 \\ 2 \\ 3 \end{bmatrix} * \begin{bmatrix}3 \\ 2 \\ 1 \end{bmatrix} = 3 + 4 + 3 = 10 $$

Matrix Multiplication

It's just the dot product for each vector in the matrices.

o = tf.constant([[[1,2,3]]]*3)
o
Out[191]: 
<tf.Tensor: id=11842, shape=(3, 1, 3), dtype=int32, numpy=
array([[[1, 2, 3]],
       [[1, 2, 3]],
       [[1, 2, 3]]], dtype=int32)>
t = tf.constant([[[3]]]*3)
t
Out[193]: 
<tf.Tensor: id=12168, shape=(3, 1, 1), dtype=int32, numpy=
array([[[3]],
       [[3]],
       [[3]]], dtype=int32)>

tf.matmul(t, o)
Out[195]: 
<tf.Tensor: id=12506, shape=(3, 1, 3), dtype=int32, numpy=
array([[[3, 6, 9]],
       [[3, 6, 9]],
       [[3, 6, 9]]], dtype=int32)>

Cross Product

a x b = -b x a X x X = 0 Orthogonal to initial vectors

Basis Vectors

Linearly independent vectors in a vector space that, when linearly combined, makes up all of the other vectors in the space. In more general terms, a basis is a linearly independent spanning set.

Unit Vectors

In a normed vector space is a vector of length 1. Unit Vectors pointing x, y, z =

$$ \hat{i} = (1, 0, 0), \hat{j} = (0, 1, 0), \hat{k} = (0, 0, 1) $$

$$ 4\hat{i} = (4, 0, 0) $$

Parallel Vectors

Two vectors are parallel if one is a scalar multiple of the other.

$$ \vec{x} {\parallel } 2\vec{x} {\parallel } 0.5\vec{x} $$

Orthorgonal Vectors

When dot-product of two different vectors is close to 0.

Two vectors which are orthogonal and of length 1 are said to be orthonormal.

Orthogonality is the generalization of the notion of perpendicularity.

Matrix Inverse

$$ X^{-1} * X = Identity $$

The transpose AT of an m×n matrix A is the n×m matrix whose (i,j)-entry is aji.

Scaling

$$ A = \begin{bmatrix} a & b \\\\ c & d \\\\ \end{bmatrix} $$

Shear

$$ A = \begin{bmatrix} 1 & a \\\\ 0 & 1 \\\\ \end{bmatrix} $$

Rotation

R =

P = radius cos alpha, r sin alpha P' = r cos(alpha + theta), r sin(alpha + theta)

Orthographic

Cuboid

Like cube, but doesn't have to be all equal.

Formal Linear Properties

$$ L(\vec{v} + \vec{w}) = L(\vec{v}) + L(\vec{w}) $$

$$ L(c\vec{v}) = cL(\vec{v}) $$

Linear transformation is a transformation that preserves spacing. For example a projects of dots in 2D with space 1 projected onto 1D still have spacing of 1.

Singular Value Decomposition SVD

Matrix where vectors are orthogonal.

![SVD] (https://research.fb.com/wp-content/uploads/2016/11/post00049_image0001.png) Facebook Fast Randomized SVD

Determinant

How much a transformation scales a matrix by. If negative, "flips" the axis then scales. When the determinant is 0 the matrix is not invertible. Meaning there is no matrix to multiply it by to give the identity.

2x2 example.

$$ A = \begin{bmatrix} a & b \\\\ c & d \\\\ \end{bmatrix} $$

$$ \det {A} = ad - bc $$

import sympy as sp
a, b, c, d = sp.symbols('a,b,c,d')
A = sp.Matrix([[a,b], [c,d]])
A.det()

# Print what this would look like in latex
sp.latex(A)
sp.latex(A.det())

Eigenvalues and Eigenvectors

A scalar $\lambda$ (eigenvalue) when multiplied by an eigenvector is equal to the original matrix A multiplied by the eigenvector.

$$ A \vec{x} = \lambda\vec{x} $$

Find

on $\det(A - \lambda I_3) = 0$, i.e.

Linear Combination

Two or more vectors with scalar weights multipled and added.

$$ Ax + By = C $$

$$ A = \begin{bmatrix} 1 & x & x^2 \\\\\\ 1 & y & y^2 \\\\\\ 1 & z & z^2 \end{bmatrix} x = 2 B = \begin{bmatrix} 1 & x & x^2 \\\\\\ 1 & y & y^2 \\\\\\ 1 & z & z^2 \end{bmatrix} y = 3 $$

The Frobenius norm

$$ ||A||_F = [\\sum_{i,j} abs(a_{i,j})^2]^{1/2} $$