Subspaces

Sets that are closed under the two allowed operations.

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

What this lesson teaches

The student tests whether a set is a subspace and identifies the column space and null space.

A subspace contains the zero vector and is closed under addition and scaling.
In the plane the only subspaces are the origin, a line through it, and the whole plane.
The column space is what A can reach; the null space is what A crushes to zero.

Straightforward practice. Get the method working first.

5 problems

The line y = −4x. Is it a subspace? 1 yes, 0 no.
Answer 1
Why Yes.
What is the null space of A?
Answer Every vector that A sends to the zero vector.
Why The solutions of Ax = 0.
The set containing only the zero vector. Is it a subspace? 1 yes, 0 no.
Answer 1
Why It passes all three tests trivially.
The whole plane. Is it a subspace of itself? 1 yes, 0 no.
Answer 1
Why Nothing escapes it.
A disc of radius 1 about the origin. Is it a subspace? 1 yes, 0 no.
Answer 0
Why Scaling a vector up leaves the disc.

The same ideas with more to keep track of.

3 problems

How many subspaces of the plane have dimension 2?
Answer 1
Why Only the whole plane itself.
A matrix whose columns span the plane. What is the dimension of its column space?
Answer 2
Why The whole plane.
An invertible matrix. What is the dimension of its null space?
Answer 0
Why Only the zero vector is crushed.

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.
Answer A subspace: The line y = 2x, The origin alone · Not a subspace: The line y = 2x + 3, The first quadrant
Why The first quadrant is not closed under scaling by −1.
Does every subspace contain the zero vector? 1 yes, 0 no.
Answer 1
Why Scaling any member by 0.
The Line: The line y = 3x. Is it a subspace of the plane? 1 yes, 0 no.
Answer 1
Why Yes.
The Shifted Line: The line y = 3x + 5. Is it a subspace? 1 yes, 0 no.
Answer 0
Why No.