Basis and Dimension

The smallest set that still reaches everything.

12 problems · about 23 minutes · N-VM.B.4, N-VM.B.5

What this lesson teaches

The student identifies a basis and determines the dimension of a subspace.

A basis spans the space and is independent: it reaches everything with nothing to spare.
Every basis of a space has the same size, and that number is the dimension.
The weights on a basis are a vector’s coordinates in that basis.

Straightforward practice. Get the method working first.

5 problems

A basis of four-dimensional space. How many vectors does it contain?
Answer 4
Why 4.
What two things must a basis do?
Answer Span the space and be linearly independent.
Why Spanning and independent.
⟨1, 0⟩ and ⟨0, 1⟩. Do they form a basis of the plane? 1 yes, 0 no.
Answer 1
Why Independent and spanning.
⟨2, 4⟩ and ⟨1, 2⟩. Do they form a basis of the plane? 1 yes, 0 no.
Answer 0
Why Dependent.
Three vectors in the plane. Can they be a basis of it? 1 yes, 0 no.
Answer 0
Why Too many to be independent.

The same ideas with more to keep track of.

3 problems

One vector in the plane. Can it be a basis of the whole plane? 1 yes, 0 no.
Answer 0
Why It cannot span.
The dimension of the whole plane. What is it?
Answer 2
Why Two independent directions.
A vector written as 3⟨1, 0⟩ + 5⟨0, 1⟩. What is its second coordinate in that basis?
Answer 5
Why The weight on the second basis vector.

Mixed problems. Work out what kind of question it is before you start.

4 problems

Sort each set by whether it is a basis of the plane.
Answer A basis of the plane: ⟨1, 0⟩ and ⟨0, 1⟩, ⟨1, 1⟩ and ⟨1, −1⟩ · Not a basis: ⟨1, 2⟩ and ⟨2, 4⟩, ⟨1, 0⟩ alone
Why Check the count first, then independence.
The dimension of the space containing only the zero vector. What is it?
Answer 0
Why No vectors are needed.
The Count: A basis of three-dimensional space. How many vectors does it contain?
Answer 3
Why 3.
The Line: A line through the origin in the plane. What is its dimension?
Answer 1
Why 1.