Lesson 2-3: Vector Spaces & Transformations

Vector Spaces & Transformations

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

Learning Objectives

Vector Spaces

Understand axioms, subspaces, span, and linear independence

Basis & Dimension

Build bases and compute dimension as size of any basis

Linear Transformations

Represent with matrices relative to bases; understand composition

Geometric Interpretation

Interpret matrices as rotations, scalings, reflections, and shears

Core Knowledge Points

Span and Linear Independence

A set spans a space if their linear combinations cover the space; it is independent if the only solution to $c_1v_1+cdots+c_kv_k=0$ is all $c_i=0$ .

Basis and Coordinates

A basis is an independent spanning set. Each vector has unique coordinates relative to a basis; change of basis is performed by an invertible matrix.

Linear Transformations and Matrices

A linear map T is determined by images of basis vectors; the matrix of T sends coordinate vectors accordingly. Composition of linear maps corresponds to matrix multiplication.

Worked Examples

Build a Basis and Coordinates

In R², basis $\{(1,0),(1,1)\}$ . Find coordinates of v=(3,4).

Solve $a(1,0)+b(1,1)=(3,4)$ → a=3-b, b=4 → a=-1. So [v]_B = (-1,4).

Linear Map Matrix

Let T rotate by θ and scale by s: in standard basis, $A=s\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$ .

Practice Problems

Problem 1: Independence Test

Determine whether $\{(1,2,0),(0,1,1),(1,3,1)\}$ is independent in R³.

Problem 2: Basis Construction

Extend $\{(1,0,0),(0,1,0)\}$ to a basis of R³ and find coordinates of $(2,1,3)$ .

Problem 3: Matrix of a Transformation

Find the matrix for reflection across y=x in R² and describe its geometric effect.

Key Takeaways

Vector Spaces

Span and independence define structure; bases coordinate vectors

Dimension

All bases of a finite-dimensional space have equal size

Linear Maps

Matrices represent maps; composition ↔ multiplication

Geometry

Rotations, scalings, reflections, shears are linear transformations

Advanced Topics

Subspaces, Column Space, Null Space

Column space: span of columns, describes reachable vectors b in Ax=b
Null space: solutions of Ax=0, encodes non-uniqueness and constraints
Rank-nullity: dim(col A) + dim(null A) = n

Linear Maps as Geometry

Scaling stretches distances along axes
Rotation preserves lengths/orientation magnitude but changes directions
Shear slides layers parallel to an axis

Eigen-Intuition (Preview)

Directions kept by a transformation up to scaling. Useful to understand repeated applications of a map.

Change of Basis

Matrix representation depends on chosen basis; similar matrices represent the same linear map under different coordinates.

Strategy Map

Identify candidate subspaces (span, solution sets, images)
Test subspace criteria: 0 in set; closed under + and scalar ×
Build a basis via independence tests or row-reduction
Represent linear maps by matrices under chosen bases
Interpret composition through matrix multiplication

Additional Practice

Set E: Subspaces

E1. Determine if S= $\{(x,y,z) : x+y+z=0\}$ is a subspace of R³
E2. Find a basis for S and its dimension
E3. Express an arbitrary vector in S in terms of your basis

Set F: Images & Kernels

F1. For A (3×3), compute basis of col(A) and null(A)
F2. Verify rank-nullity relation
F3. Describe the geometry of col(A) in R³

FAQ

How to determine if a set of vectors forms a basis?

For a target space of dimension d, find d linearly independent vectors that span the space.

Does changing basis change the linear transformation?

No, it doesn't change the "transformation itself", but changes its matrix representation; similar matrices represent the same linear transformation.

What is the intuitive meaning of null space and column space?

Null space represents directions mapped to 0; column space represents all reachable output directions.

Project Ideas

2D Geometry Tool: Implement combinations of rotation, scaling, and shear matrices, observe shape changes
Coordinate Transformation Visualization: Observe the same vector's coordinates under different bases
Column Space/Null Space Explorer: Automatically display basis and dimension through RREF

Summary Checklist

[ ] I can determine if a set is a subspace
[ ] I can construct bases and compute dimensions
[ ] I can give the matrix of a linear transformation under a given basis
[ ] I can explain column space, null space, and their relationship

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