Example 1: Vector Addition and Scalar Multiplication

Problem: Given v = [2, -1, 3] and w = [1, 4, -2], find 2v + 3w.

Solution:
First, calculate 2v: 2[2, -1, 3] = [4, -2, 6]
Then, calculate 3w: 3[1, 4, -2] = [3, 12, -6]
Add: [4, -2, 6] + [3, 12, -6] = [7, 10, 0]

Example 2: Dot Product

Problem: Find the dot product of u = [3, 4] and v = [1, 2].

Solution:
u · v = (3)(1) + (4)(2) = 3 + 8 = 11

The dot product measures how aligned two vectors are. If the dot product is zero, the vectors are perpendicular (orthogonal).

Example 3: Matrix Multiplication

Problem: Multiply matrix A = [[1, 2], [3, 4]] by matrix B = [[5, 6], [7, 8]].

Solution:
First row, first column: (1)(5) + (2)(7) = 5 + 14 = 19
First row, second column: (1)(6) + (2)(8) = 6 + 16 = 22
Second row, first column: (3)(5) + (4)(7) = 15 + 28 = 43
Second row, second column: (3)(6) + (4)(8) = 18 + 32 = 50
Result: AB = [[19, 22], [43, 50]]

Example 4: Solving a System of Equations

Problem: Solve the system: 2x + y = 7, x - y = 2

Solution using substitution:
From the second equation: x = y + 2
Substitute into the first: 2(y + 2) + y = 7
2y + 4 + y = 7
3y = 3, so y = 1
Then x = 1 + 2 = 3
Solution: (x, y) = (3, 1)

In matrix form: [[2, 1], [1, -1]] · [x, y] = [7, 2]. This can also be solved using matrix inverse or Gaussian elimination.

Check Matrix Dimensions

Before multiplying matrices, verify that dimensions are compatible: an m×n matrix can multiply an n×p matrix (the inner dimensions must match), resulting in an m×p matrix. For addition or subtraction, matrices must have exactly the same dimensions. This is the most common source of errors in linear algebra calculations.

Visualize in Lower Dimensions

Even if working in higher dimensions, visualize concepts using 2D or 3D examples first. Draw vectors as arrows, understand transformations geometrically, and see how matrix operations affect space. This geometric intuition helps you understand abstract concepts and catch errors that don't make geometric sense.

Organize Your Work

Matrix and vector calculations involve many numbers. Write them clearly, align columns properly, and work systematically row by row or column by column. Use parentheses and brackets consistently. Label intermediate results. Organized work prevents arithmetic errors and makes it easier to find mistakes when checking your answers.

Master the Fundamentals

Linear algebra builds on itself. Make sure you understand vector operations before moving to matrices, and understand matrix operations before tackling determinants, eigenvalues, and advanced topics. Each concept provides the foundation for the next, so gaps in understanding compound quickly.

Vector

An ordered list of numbers representing a point or direction in space. Vectors have magnitude (length) and direction. Written as [x, y] in 2D or [x, y, z] in 3D. Vectors are the building blocks of linear algebra.

Matrix

A rectangular array of numbers organized in rows and columns. An m×n matrix has m rows and n columns. Matrices represent linear transformations, systems of equations, and data. They're fundamental objects in linear algebra.

Linear Combination

An expression formed by multiplying vectors by scalars and adding the results. For example, 3v + 2w - 4u is a linear combination of vectors v, w, and u. Linear combinations are central to understanding vector spaces.

Dot Product

Also called scalar product or inner product, it's an operation that takes two vectors and returns a scalar. For vectors u and v, u · v = u₁v₁ + u₂v₂ + ... The dot product measures how aligned vectors are and equals zero for perpendicular vectors.

Linear Independence

A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. Linearly independent vectors point in truly different directions and form the basis for understanding dimension.

Span

The span of a set of vectors is the collection of all possible linear combinations of those vectors. It represents all the "space" that can be reached using those vectors. Two non-parallel vectors in 2D span the entire 2D plane.

Identity Matrix

A square matrix with 1s on the main diagonal and 0s elsewhere, often denoted I. The identity matrix acts like the number 1 in matrix multiplication: AI = IA = A for any matrix A. It represents the "do nothing" transformation.

Inverse Matrix

For a square matrix A, its inverse A⁻¹ (if it exists) satisfies AA⁻¹ = A⁻¹A = I. Not all matrices have inverses—only those with non-zero determinants. The inverse "undoes" the transformation represented by the matrix.