Subspaces

Sets that are closed under the two allowed operations.

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

NameDate

Figure — use these to answer the problems

When the parallelogram survives, the column space is the whole plane. Collapse it and the column space shrinks to the line the arrows share.

first column, top2

second column, top4

first column, bottom1

second column, bottom2

Figure 1

Straightforward practice. Get the method working first.

5 problems

The line y = −4x. Is it a subspace? 1 yes, 0 no.
Answer
2
What is the null space of A?
- Every vector that A sends to the zero vector.
- The span of the columns of A.
The set containing only the zero vector. Is it a subspace? 1 yes, 0 no.
Answer
The whole plane. Is it a subspace of itself? 1 yes, 0 no.
Answer
A disc of radius 1 about the origin. Is it a subspace? 1 yes, 0 no.
Answer

The same ideas with more to keep track of.

3 problems

How many subspaces of the plane have dimension 2?
Answer
A matrix whose columns span the plane. What is the dimension of its column space?
Answer
An invertible matrix. What is the dimension of its null space?
Answer

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

4 problems

Sort each set by whether it is a subspace of the plane.
Write each item under the heading it belongs to: The line y = 2x · The line y = 2x + 3 · The origin alone · The first quadrant
A subspace
Not a subspace
Does every subspace contain the zero vector? 1 yes, 0 no.
Answer
The Line
The line y = 3x. Is it a subspace of the plane? 1 yes, 0 no.
Answer
The Shifted Line
The line y = 3x + 5. Is it a subspace? 1 yes, 0 no.
Answer