Vector Spaces & Linear Transformations

Understand the structure of vector spaces, choose bases to describe coordinates, compute dimensions, and study linear transformations through their matrix representations and geometric actions.

12th Grade

Linear Algebra

~90 min

🎮 Interactive Activity: Basis Checker

Determine if the vectors form a basis for R²!

Vectors:

v₁ = [1, 0]

v₂ = [0, 1]

Do they form a basis?

🎮 Interactive Activity: Linear Transformation Applier

Apply the linear transformation to the vector!

Transformation Matrix:

[[1, 0], [0, 1]]

applied to

Vector:

[2, 3]

Result: [x, y] = ?

1. Vector Spaces and Axioms

Understanding Vector Spaces

A vector space is a set V with two operations (addition and scalar multiplication) that satisfy eight axioms: closure, associativity, commutativity, identity, inverse, and distributive laws.

Example 1: R² as a Vector Space

Elements: All ordered pairs (x, y) where $x, y \\in \\mathbb{R}$

Addition: (x₁, y₁) + (x₂, y₂) = (x₁ + x₂, y₁ + y₂)

Scalar multiplication: k(x, y) = (kx, ky)

Zero vector: (0, 0)

Verification: All axioms are satisfied

Example 2: Subspaces

Definition: A subset W of V is a subspace if it's closed under addition and scalar multiplication

Example: The line y = 2x in R² is a 1-dimensional subspace

Basis: $\\{(1, 2)\\}$ spans this subspace

Dimension: 1

Example 3: Polynomial Vector Space

Space: P₂ = $\\{ax^2 + bx + c : a, b, c \\in \\mathbb{R}\\}$

Operations: Standard polynomial addition and scalar multiplication

Basis: $\\{1, x, x^2\\}$

Dimension: 3

2. Span and Linear Independence

3. Basis and Coordinates

4. Dimension of Vector Spaces

5. Linear Transformations

6. Matrix Representation of Linear Transformations

7. Geometric Interpretations

Frequently Asked Questions

Practice Time!

Practice Quiz

Questions

Correct

Score

What is the dimension of R³?

○

Which set of vectors forms a basis for R²?

○

What is the span of vectors {(1,0), (0,1)} in R²?

○

If T is a linear transformation with matrix [[1, 2], [3, 4]], what is T(1, 0)?

○

What does it mean for vectors to be linearly independent?

○

What is the dimension of the span of {(1,2), (2,4)} in R²?

○

If a linear transformation T: R² → R² has matrix [[cos θ, -sin θ], [sin θ, cos θ]], what does it represent?

○

What is the null space of a matrix?

○

If a vector space has dimension n, how many vectors are in any basis?

○

What is the rank of a matrix?

○