Basis and Dimension

The smallest set that still reaches everything.

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

NameDate

Figure — use these to answer the problems

These two arrows are a candidate basis. While the determinant is non-zero they span the plane; collapse it and they no longer do.

first vector, top1

second vector, top-1

first vector, bottom1

second vector, bottom1

Figure 1

Straightforward practice. Get the method working first.

5 problems

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

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
The dimension of the whole plane. What is it?
Answer
A vector written as 3⟨1, 0⟩ + 5⟨0, 1⟩. What is its second coordinate in that basis?
Answer

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.
Write each item under the heading it belongs to: ⟨1, 0⟩ and ⟨0, 1⟩ · ⟨1, 1⟩ and ⟨1, −1⟩ · ⟨1, 2⟩ and ⟨2, 4⟩ · ⟨1, 0⟩ alone
A basis of the plane
Not a basis
The dimension of the space containing only the zero vector. What is it?
Answer
The Count
A basis of three-dimensional space. How many vectors does it contain?
Answer
The Line
A line through the origin in the plane. What is its dimension?
Answer