Mathematical object

A mathematical object is an abstract concept which can be a value that can be assigned to a symbol, and therefore can be involved in formulas.

• Examples numbers, expressions, shapes, functions, and sets.
• Complex Objects: theorems, proofs.

Tensor

Algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space.

• Latin: tendere meaning 'to stretch'

Scaler($s \in \mathbb{R}$)

Scalars are real numbers used in linear algebra

• A single number, a 0-dimensional tensor.
• Example: $5$, $-3.14$, $\pi$, $e$

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Matrices($A \in \mathbb{R}^{n \times m}$)

A matrix is a 2D array of numbers or table of numbers.

$A = \begin{bmatrix} 85 & 76 & 66 & 5 \\ 94 & 75 & 18 & 28 \\ 68 & 40 & 71 & 5 \end{bmatrix}$

Representation:

In mathematics:

$A \in \mathbb{R}^{3 \times 4}$

• Where $A$ is a real-valued matrix with 4 rows and 2 columns.

In programming:

A = np.array([[85, 76, 66, 5],
              [94, 75, 18, 28],
              [68, 40, 71, 5]])

In theory:

• Uppercase letters (A, B, X) → Matrices
• Lowercase letters (x, y, z) → Vectors or scalars

Dimension:

$(m \times n)$

Where:

• $m$ = rows
• $n$ = columns
• Example $3\times4$ matrix in the example

Square Matrix

• A matrix with the same number of rows and columns ($m = n$).

Element notation:

$A_{ij}$

• The element in the $i-th$ row and $j-th$ column.

Example:

• $A_{11}$ = 85 → Row 1, Column 1
• $A_{32} = 40$ → Row 3, Column 2
• $A_{41} = 5$ → Row 4, Column 1
• $A_{23} = 18$ → Row 2, Column 3
• $A_{64} = undefined$ → 6th row, 4th column does not exist

Use in Machine Learning

Represents Data Matrix, Model Parameters, Transformations

If we have:

• $m$ training examples
• $n$ features

The data matrix is:

$$X = \begin{bmatrix} --- x^{(1)} --- \\ --- x^{(2)} --- \\ \vdots \\ --- x^{(m)} --- \end{bmatrix}$$

Dimension $m \times n$ where:

• Each row = one training example
• Each column = one feature

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Vectors($\vec{x}$)

A vector is a Matrix with 1 Column

• Represents A point in $n$ high-dimensional space
• Latin: vector , meaning "carrier" or "driver"
• Have A direction ($\vec{x}$) & A magnitude

$x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$

Represent as:

In Maths

$y \in \mathbb{R}^4$

In Programming

y = np.array([460, 232, 315, 178])

In Theory:

• Uppercase letters (A, B, X) → Matrices

Dimension: $n \times 1$

In ML, vectors represent:

• A data point → a vector
• A feature column → a direction
• A model weight vector → a direction of best fit

Example

$y = \begin{bmatrix} 460 \\ 232 \\ 315 \\ 178 \end{bmatrix}$

• 4 × 1 matrix Or a 4-dimensional vector

Element Indexing

$y_i$ = i-th element.

• In mathematics, indexing usually starts at 1.
• In programming indexing often starts at 0.
• Unless otherwise specified, assume one-indexed notation in linear algebra.

Example:

• $y_1 = 460$
• $y_2 = 232$
• $y_3 = 315$
• $y_4 = 178$

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Transpose ($\mathbf{x}^T$)

Transpose swaps rows and columns.

If: $A \in \mathbb{R}^{m \times n}$ then: $A^T \in \mathbb{R}^{n \times m}$

Element-wise:

$(A^T)_{ij} = A_{ji}$

• A column vector becomes a row vector.

Given:

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

Then:

$$A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}$$

Used heavily in:

• Normal Equation
• Gradient derivations

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Identity Matrix ($I$)

The identity matrix is the matrix equivalent of the number 1.

It is a square matrix with:

• 1’s on the diagonal
• 0’s everywhere else

$$I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$$

Property:

$$AI = IA = A$$

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Inverse Matrix($A^{-1}$)

The inverse of a matrix is like division.

• Only square matrices can have inverses.

Matrix inverse satisfies:

$$A^{-1}A = AA^{-1} = I$$

Used in Normal Equation:

$$\theta = (X^T X)^{-1} X^T y$$

Not all square matrices are invertible.

1. Invertible/ non-singular Matrix

A matrix can be inverted

• it has an inverse if it is full rank (rows and columns are linearly independent).

2. Non-Invertible/ Singular Matrix/ Degenerate Matrix

A matrix that does not have an inverse

• Does not have a inverse because it is not full rank (rows or columns are linearly dependent).

Cause for non invertible Matrix:

• Redundant feature: two feature related by a linear equation x2 = kx1 eg: size in feet and meter
• More feature than training set(m<=n)): delete some feature or use regularization

Octave method for inverting matrix:

• pinv(A) : Pseudo Inverse, calculates inverse even if matrix is non invertible
• inv(A) : Inverse

Determinant ( $det(A)$)

The determinant tells us whether a matrix is invertible.

For a 2 × 2 matrix:

$$A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$ $$\det(A) = ad - bc$$

If: $det(A) \neq 0$: The matrix is invertible.

If: $\det(A) = 0$ : The matrix is singular (not invertible).

• Either no solution
• Or infinitely many solution

Use in Machine Learning:

• Normal Equation requires matrix inversion.

Closed-form solution:

$$\theta = (X^T X)^{-1} X^T y$$

• In practice, we use numerical methods to avoid instability of matrix inversion.
• Regularization can help make matrices invertible by adding a small value to the diagonal (Ridge Regression).

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A matrix is a transformation of space.

All machine learning models are compositions of transformations.

If:

$$y = Ax$$

Then $A$ transforms vector $x$ into a new vector $y$.

Geometrically, a matrix can:

• Stretch
• Compress
• Rotate
• Reflect
• Shear
• Project

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Matrix Addition/Subtraction

When Is Addition Allowed?

Addition is done element by element.

Two matrices can be added only if they have the same dimensions.

If:

$A, B \in \mathbb{R}^{m \times n}$

then:

$C = A + B$ where $(A + B)_{ij} = A_{ij} + B_{ij}$

is also an $( m \times n$) matrix.

$$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$$ $$A + B = \begin{bmatrix} 1+5 & 2+6 \\ 3+7 & 4+8 \end{bmatrix} = \begin{bmatrix} 6 & 8 \\ 10 & 12 \end{bmatrix}$$

Subtraction is the same but with minus signs.

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Scalar Multiplication / Division

Scalar multiplication is multiplying every element of a matrix by a single number (scalar).

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Matrix-Matrix Multiplication

Element-wise: $C_{ij} = A_{ik}B_{kj}$ (sum over k)

Given 2 Matrices:

• $A$ is $(m \times n)$ or $A \in \mathbb{R}^{m \times n}$
• $B$ is $(n \times p)$ or $B \in \mathbb{R}^{n \times p}$

Then:

$$C = AB$$

where $C$ is a new matrix with dimensions:

• $C(m \times p)$ or $C\in \mathbb{R}^{m \times p}$
• inner dimensions must match (n) : $(m \times n)(n \times p) \rightarrow (m \times p)$

Properties:

• Not commutative: $AB \ne BA$: Order matters
• Associative: $(AB)C = A(BC)$

Use in Machine Learning

Everything in deep learning is matrix multiplication:

• Inputs × Weights
• Weights × Activations
• Gradient updates

Neural network forward pass: $Z = WX + b$

Backpropagation is also matrix calculus.

Understanding multivariate linear algebra makes deep learning much easier to grasp.

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Vectorization: Matrix-Vector Multiplication

If:

• $X$ is $m \times n$ Matrix
• $\theta$ is $n \times 1$ Vector

Then

$$h = X\theta$$

• Produces $h (m \times 1)$ Vector

Use in Machine Learning

• This gives predictions for all training examples in one operation.
• Faster computation: optimized hardware usage (CPU/GPU)
• Clean mathematical formulation

Linear regression hypothesis:

$$h_\theta(x) = \theta^T x$$

For all training examples:

$$h = X\theta$$

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Dot Product($a.b$)

The dot product is defined between two vectors of the same dimension.

If:

$$x, y \in \mathbb{R}^n$$

Then their dot product is:

$$x^T y$$ $$x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix} \quad y = \begin{bmatrix} y_1 \\ y_2 \\ \vdots \\ y_n \end{bmatrix}$$ $$x^T y = x_1 y_1 + x_2 y_2 + \dots + x_n y_n$$

It produces a single number (a scalar).

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Summary

Key ideas:

• Vectors represent features and parameters
• Matrices represent datasets
• Matrix multiplication enables fast prediction
• Transpose and inverse enable optimization
• Vectorization is essential for performance