Quick Linear Algebra

Quick Linear Algebra#

Welcome to Quick Linear Algebra, where we uncover the math behind all the buzzwords — without turning your notebook into an ancient scroll of equations.

If you’ve ever thought,

“Why are data scientists obsessed with matrices?”

the answer is simple: because Excel walked so linear algebra could run. 🏃‍♂️💨

🧮 Why Linear Algebra Matters for Business ML#

Machine learning is basically “math applied to big tables.”

Linear algebra is the grammar for working with those tables — helping models:

  • Combine customer data 🧾

  • Transform variables 📈

  • Make predictions 🔮

In short: Linear algebra = data manipulation with style.

📊 Meet the Stars: Scalars, Vectors, and Matrices#

Term

Symbol

Think Of It As

Business Example

Scalar

( x )

A single number

One product’s price

Vector

( \vec{x} )

A list of numbers

Prices of all your products

Matrix

( A )

A table of numbers

Product prices × stores

Tensor

( \mathcal{T} )

Multi-dimensional array

Price × store × day

So yeah, matrices are just glorified spreadsheets — except they actually behave when you multiply them.

💡 Visualizing the Concept#

You See This In Excel

ML Sees This As

Columns of “Customer ID,” “Spend,” “Region”

Feature Matrix ( X )

Column of “Churned = Yes/No”

Target Vector ( y )

Every ML model does something like this:

[ \hat{y} = X \cdot \beta ]

Where:

  • ( X ) → all your features (inputs)

  • ( \beta ) → learned weights (importance of each feature)

  • ( \hat{y} ) → predicted outcome

📊 Translation: “Combine all business variables with their importance weights → get your prediction.”

⚙️ Core Operations (Without the Fear)#

Operation

Math Form

Plain English

Business Analogy

Addition

( A + B )

Combine datasets

Merge sales from two regions

Multiplication

( A \times B )

Weighted sum

Apply importance to each factor

Dot Product

( \vec{x} \cdot \vec{y} )

Measure alignment

How similar two customers’ behaviors are

Transpose

( A^T )

Flip rows ↔ columns

Switch from “per store” view to “per product” view

Inverse

( A^{-1} )

Undo a transformation

Back out discounts to get list price

Identity Matrix

( I )

Do nothing (neutral)

Like “no change” in a KPI dashboard

🧩 Practice Corner #1: “The Matrix Manager”#

You’re analyzing product data:

[ X = \begin{bmatrix} \text{AdSpend} & \text{Discount}
10 & 5
8 & 7 \end{bmatrix}, \quad \beta = \begin{bmatrix} 2 \ 3 \end{bmatrix} ]

Compute ( X \cdot \beta ).

🧠 Hint: Multiply row-by-row and add the results.

Row

Calculation

Result

1

(10×2) + (5×3)

35

2

(8×2) + (7×3)

37

✅ So predicted sales = [35, 37]. Boom. You just did linear algebra — and a regression prediction.

📦 Why Businesses Should Care#

Because every model doing:

  • Forecasting,

  • Segmentation, or

  • Optimization

is secretly crunching matrix math behind the scenes.

ML Task

Linear Algebra Role

Linear Regression

Combine feature matrix with coefficients

PCA / Dimensionality Reduction

Rotate data into simpler shapes

Recommendations

Compute similarity between users/items

Deep Learning

Multiply enormous matrices really, really fast

TL;DR: If data is the new oil, matrices are the refineries.

🧩 Practice Corner #2: “Matrix or Mayhem?”#

Decide whether each business example uses linear algebra (✅) or not (❌):

Scenario

Linear Algebra?

1. Adding total monthly sales from all stores

❌ (simple sum)

2. Predicting sales using ad spend and discounts

✅

3. Comparing customer segments via similarity scores

✅

4. Counting how many customers churned last quarter

❌

📘 Quick Recap#

✅ Scalars → single numbers ✅ Vectors → columns of data ✅ Matrices → multi-feature tables ✅ Dot products → similarity or weighted sums ✅ Linear algebra → Excel formulas with superhero capes

🧭 Up Next#

Next stop: Calculus Essentials → We’ll see how models “learn” — by using calculus to reduce their sadness (loss) one tiny derivative at a time. 💧📉

Notations of Basic Algebra#

1. Number:

  • A simple number: \(5\)

  • A negative number: \(-3\)

  • A decimal number: \(2.718\)

  • A fraction: \(\frac{1}{2}\)

2. Vector:

  • A column vector: \(\mathbf{v} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}\)

  • A row vector: \(\mathbf{u} = \begin{bmatrix} a & b & c \end{bmatrix}\)

3. Dot Product:

  • The dot product of two vectors \(\mathbf{a}\) and \(\mathbf{b}\): \(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + \cdots + a_nb_n\)

  • Example with specific vectors: \(\begin{bmatrix} 1 \\ 2 \end{bmatrix} \cdot \begin{bmatrix} 3 \\ 4 \end{bmatrix} = (1 \times 3) + (2 \times 4) = 3 + 8 = 11\)

4. Multiplication:

  • Scalar multiplication: \(3 \times 4 = 12\) or \(3 \cdot 4 = 12\)

  • Matrix-vector multiplication: \(A \mathbf{x} = \mathbf{b}\), where \(A\) is a matrix and \(\mathbf{x}, \mathbf{b}\) are vectors. Example: \(\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 5 \\ 6 \end{bmatrix} = \begin{bmatrix} (1 \times 5) + (2 \times 6) \\ (3 \times 5) + (4 \times 6) \end{bmatrix} = \begin{bmatrix} 17 \\ 39 \end{bmatrix}\)

  • Matrix-matrix multiplication: \(AB = C\) Example: \(\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} = \begin{bmatrix} (1 \times 2) + (0 \times 4) & (1 \times 3) + (0 \times 5) \\ (0 \times 2) + (1 \times 4) & (0 \times 3) + (1 \times 5) \end{bmatrix} = \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix}\)

5. Inverse:

  • The inverse of a matrix \(A\) is denoted as \(A^{-1}\), such that \(AA^{-1} = A^{-1}A = I\), where \(I\) is the identity matrix.

  • Example of a \(2 \times 2\) inverse: If \(A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}\), then \(A^{-1} = \frac{1}{ad - bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}\), provided \(ad - bc \neq 0\). Example with numbers: If \(A = \begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix}\), then \(ad-bc = (2 \times 3) - (1 \times 4) = 6 - 4 = 2\), and \(A^{-1} = \frac{1}{2} \begin{bmatrix} 3 & -1 \\ -4 & 2 \end{bmatrix} = \begin{bmatrix} 1.5 & -0.5 \\ -2 & 1 \end{bmatrix}\).

6. Identity Matrix:

  • A \(2 \times 2\) identity matrix: \(I_2 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}\)

  • A \(3 \times 3\) identity matrix: \(I_3 = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}\)

7. Identity and Inverse Multiplication:

  • Multiplication of a matrix by its inverse results in the identity matrix: \(AA^{-1} = I\) Example: \(\begin{bmatrix} 2 & 1 \\ 4 & 3 \end{bmatrix} \begin{bmatrix} 1.5 & -0.5 \\ -2 & 1 \end{bmatrix} = \begin{bmatrix} (2 \times 1.5) + (1 \times -2) & (2 \times -0.5) + (1 \times 1) \\ (4 \times 1.5) + (3 \times -2) & (4 \times -0.5) + (3 \times 1) \end{bmatrix} = \begin{bmatrix} 3 - 2 & -1 + 1 \\ 6 - 6 & -2 + 3 \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} = I\)

8. Transpose:

  • The transpose of a matrix \(A\) is denoted as \(A^T\) or \(A'\). The rows of \(A\) become the columns of \(A^T\), and vice versa.

  • Example: If \(A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}\), then \(A^T = \begin{bmatrix} 1 & 4 \\ 2 & 5 \\ 3 & 6 \end{bmatrix}\).

  • Transpose of a vector: If \(\mathbf{v} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}\), then \(\mathbf{v}^T = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}\).

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