LA 101
M01 · L02
Introduction

Scalars, Vectors & Matrices

The three fundamental objects of linear algebra. Every equation, every transformation, every algorithm in this subject is built from scalars, vectors, and matrices. Let us meet them one by one.

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LA 101
M01 · L02
Building block 1

Scalars

A scalar is just a single number. It has magnitude only — no direction. Temperature, mass, speed: all scalars. The name comes from "scale" because scalars scale other objects up or down.

Has
Magnitude
No
Direction
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LA 101
M01 · L02
Building block 2

Vectors

An ordered list of numbers — components that describe direction and magnitude. In 2D, a vector has two entries. In 3D, three. In machine learning, vectors can have millions.

Vector notation
\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ v_3 \end{bmatrix}
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LA 101
M01 · L02
Operations

Vector Operations

Two fundamental operations define vector algebra: addition (component by component) and scalar multiplication (stretch or shrink by a number).

Key insight
These two operations are the foundation of all linear algebra. Every vector in a space can be built from them.
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LA 101
M01 · L02
Building block 3

Matrices

A rectangular array of numbers with m rows and n columns. Far more than a table — matrices represent transformations, systems of equations, and datasets.

m × n matrix
A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{bmatrix}
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LA 101
M01 · L02
Notation

Matrix Elements

We identify each entry by its row and column. The element a_ij sits in row i, column j. Dimensions are always written as rows × columns.

Entry
aij
Dimensions
m × n
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LA 101
M01 · L02
Special matrix

The Identity Matrix

Ones on the diagonal, zeros everywhere else. Multiplying by the identity changes nothing — like multiplying a number by 1.

Identity matrix
I = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}
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LA 101
M01 · L02
Gallery

Special Matrices

Several matrix types appear constantly in science and engineering.

  • Zero matrix — all entries are zero
  • Diagonal — nonzero only on the main diagonal
  • Symmetric — equals its own transpose
  • Identity — the "do nothing" transformation
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LA 101
M01 · L02
Operations

Matrix Operations

Matrices can be added element by element (same dimensions required) and scaled by a constant. These mirror the operations we saw for vectors.

Rule
Matrix addition: (A+B)_ij = A_ij + B_ij. Scalar multiplication: (cA)_ij = c · A_ij.
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LA 101
M01 · L02
Key operation

The Transpose

Flip a matrix over its diagonal — rows become columns, columns become rows. If A is m×n, then A^T is n×m.

Transpose example
A^T_{ij} = A_{ji}
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LA 101
M01 · L02
Impact

Why These Matter

Scalars, vectors, and matrices are the language of engineering. Every advanced topic is built from them.

  • Signals are vectors
  • Datasets are matrices
  • Forces are vectors
  • Transformations are matrices
  • Weights in neural nets are scalars in matrices
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LA 101
Knowledge Check

Check what stuck

Three questions from this lesson. Answer to see why — the explanation appears whether you were right or wrong. Nothing is scored or saved.

Question 1 of 0
Score 0/0

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LA 101
Summary
Recap

What you learned

Scalars are single numbers. Vectors are ordered lists with direction and magnitude. Matrices are rectangular arrays that encode transformations and data. Together, they form the alphabet of linear algebra.

Next Lesson
Geometric Interpretation →
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