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Maths I — Week 8: Linear Algebra

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Maths I — Week 8: Linear Algebra

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Maths I — Week 8: Linear Algebra

Short description. Linear Algebra is the mathematics of organized data. Matrices allow us to solve complex systems of equations efficiently by treating entire sets of numbers as single algebraic units.

1. Core Concept

Definition: A Matrix is a rectangular array of numbers arranged in

m

rows and

n

columns. Matrix algebra provides a deterministic framework for transformations and solving systems of linear equations.

Intuition: Imagine a matrix as a spreadsheet that can be multiplied by other spreadsheets. It represents a "transformation" of space—rotating, scaling, or flipping vectors.

Formula / Rule:

A_{m \times n} \cdot B_{n \times p} = C_{m \times p}

where the element

c_{ij}

is the dot product of the

i

-th row of

A

and the

j

-th column of

B

Important

Commutativity Fail: Unlike scalar multiplication,

AB \neq BA

in most cases. The order of multiplication is structurally sacred.

2. Pattern A — Matrix Multiplication

What to recognize: A request to find the product of two arrays

A

and

B

Abstract Solution (Strategy)

[Dimension Audit]: Verify if the number of columns in $A$ exactly matches the number of rows in $B$ .
[Dot Product Rule]: Each entry $c_{ij}$ is the sum of products of corresponding elements from Row $i$ of $A$ and Column $j$ of $B$ .
[Watch for]: Inadvertently swapping the order or incorrectly adding instead of multiplying.

Procedure

Step 1: Check dimensions ( $m \times n$ and $n \times p$ ). The outer numbers $(m, p)$ define the result size.
Step 2: For each coordinate $(i, j)$ in the result, multiply Row $i$ of $A$ with Column $j$ of $B$ element-wise and sum them.
Step 3: Repeat until the result matrix is saturated.

Worked Example:

Question: Find $AB$ for $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$ and $B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$ .

Step 1: $2 \times 2$ times $2 \times 2$ = $2 \times 2$ result.
Step 2: Top-Left coordinate $(1,1) = (1\cdot5) + (2\cdot7) = 5 + 14 = 19$ .
Step 3: Fill other coordinates similarly.
Answer: $\begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$

3. Pattern B — $2 \times 2$ Matrix Inversion

What to recognize: A question asking for

A^{-1}

where

A

is a square

2 \times 2

matrix.

Abstract Solution (Strategy)

[Determinant Check]: Calculate $\\Delta = ad - bc$ . If $\\Delta = 0$ , the inverse does not exist (Matrix is singular).
[Adjugate Swap]: Swap the main diagonal elements $(a \leftrightarrow d)$ and negate the off-diagonal elements $(-b, -c)$ .
[Scalar Scaling]: Multiply the resulting adjugate matrix by $\frac{1}{\Delta}$ .

Procedure

Step 1: Compute $D = (m_{11} \cdot m_{22}) - (m_{12} \cdot m_{21})$ .
Step 2: Rearrange: $\begin{bmatrix} a & b \\ c & d \end{bmatrix} \to \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$ .
Step 3: Multiply by $1/D$ and simplify.

Worked Example:

Question: Inverse of $\begin{bmatrix} 2 & 1 \\ 5 & 3 \end{bmatrix}$ .

Step 1: $\\det = (2\cdot3) - (1\cdot5) = 1$ .
Step 2: Swap/Negate $\to \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}$ .
Step 3: $\frac{1}{1} \cdot \begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}$ .
Answer: $\begin{bmatrix} 3 & -1 \\ -5 & 2 \end{bmatrix}$

4. Common Mistakes

Mistake	Why it happens	Correct approach
Multiplying matrices of incompatible sizes.	Ignoring the dimension audit.	Always check if Col Count of A = Row Count of B.
Forgetting to divide by the determinant in inversions.	Skipping the final scalar scaling step.	The inverse is ONLY the adjugate IF the determinant is exactly 1.
Assuming $AB = BA$ .	Applying scalar habits to matrix algebra.	The order is fundamental. $(AB)^T = B^T A^T$ .

5. Flashcards

6. Practice Targets

Attempt Maths I Graded Assignment 8.
Calculate the determinant of a $3 \times 3$ matrix using cofactor expansion on the 1st row.
Solve $Ax = b$ for $2 \times 2$ using $x = A^{-1}b$ .

7. Connections

Connects to	How
Week 2 — Polynomials	Solving systems of linear equations is the foundational step for partial fraction decomposition and curve fitting.
Data Science Core	Matrices are the primary data structure for all machine learning models (tensors).