Lesson 4: Basis and Dimension
Estimated time: 35-45 minutes
Learning Objectives
- Define a basis for a vector space
- Find a basis for the column space and null space of a matrix
- Define the dimension of a vector space
- Define rank and nullity and state the Rank Theorem
Basis
Basis: A set of vectors B = {v_1, ..., v_k} is a basis for a subspace H if: (1) B is linearly independent, and (2) H = Span(B). A basis is a minimal spanning set -- no vector is redundant.
Example
The standard basis for R^3: {e_1, e_2, e_3} = {(1,0,0), (0,1,0), (0,0,1)}. Independent and spans all of R^3.
Finding Bases from Matrices
Basis for Col(A): The pivot columns of the original matrix A form a basis for the column space.
Basis for Nul(A): Row reduce A, express free variables as parameters, and the direction vectors from the parametric form constitute a basis for the null space.
Worked Example
A = [1 2 3; 2 4 7; 1 2 4]. RREF: [1 2 0; 0 0 1; 0 0 0]. Pivots in columns 1 and 3.
Basis for Col(A): Columns 1 and 3 of the original A: {(1,2,1), (3,7,4)}.
Basis for Nul(A): Free variable x_2 = t. From RREF: x_1 = -2t, x_3 = 0. Null space basis: {(-2, 1, 0)}.
Dimension
Dimension: The dimension of a subspace H, written dim(H), is the number of vectors in any basis for H. All bases for a given subspace have the same number of vectors.
dim(R^n) = n. dim({0}) = 0. dim of a line through origin = 1. dim of a plane through origin = 2.
Rank and Nullity
Rank: rank(A) = dim(Col(A)) = number of pivot columns.
Nullity: nullity(A) = dim(Nul(A)) = number of free variables.
Rank Theorem: For an m x n matrix A: rank(A) + nullity(A) = n (number of columns).
Example
From the example above: A is 3x3. Rank = 2 (two pivots). Nullity = 1 (one free variable). 2 + 1 = 3 = n. ✓
Check Your Understanding
1. What is dim(R^5)?
2. A 4x6 matrix has rank 3. What is its nullity?
3. Can {(1,2), (2,4)} be a basis for R^2?
4. An invertible 5x5 matrix has what rank?
Key Takeaways
- Basis = linearly independent spanning set (minimal spanning, maximal independent)
- Dimension = number of vectors in any basis
- Basis for Col(A): pivot columns of the original A
- Basis for Nul(A): direction vectors from parametric form of Ax = 0
- Rank Theorem: rank + nullity = number of columns