A subspace is a set of vectors you cannot escape by adding or scaling. Doing linear algebra inside it never takes you out of it.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The three tests
It contains the zero vector, it is closed under addition, and it is closed under scaling. All three must hold.
What counts
In the plane the only subspaces are the origin alone, any line through the origin, and the whole plane.
What does not
A line missing the origin fails immediately. So does a disc, since scaling a vector up eventually leaves it.
The column space
The span of the columns of A is a subspace. It is exactly the set of vectors b for which Ax = b can be solved.
The null space
Every x with Ax = 0 forms a subspace too. It is what the transformation crushes to nothing.
Closed under both operations
A subspace is a subset that stays inside itself under addition and scalar multiplication. Those two closure conditions, plus containing zero, are the whole definition.
Every subspace contains the origin
Scaling any member by zero gives the zero vector, which must therefore be in the set. A line not through the origin is not a subspace, however linear it looks.
The important ones
The column space is the span of the columns; the null space is everything sent to zero. Both are subspaces, and the relationship between them is what the rank-nullity theorem describes.
They are flats through the origin
In three dimensions, the subspaces are the origin, lines through it, planes through it, and all of space. Knowing the complete list makes many questions answerable by inspection.
Step 2: Try It Yourself
Tap and try it out.
- i-hat lands on(2, 1)
- j-hat lands on(4, 2)
- Determinant0
The parallelogram has collapsed to a segment. The determinant is zero, the whole plane has been squashed onto a line, and no inverse can undo it.
Step 3: Watch an Example
One step at a time.
Watch Tara Reject a Set
Tara tests whether the line y = x + 1 is a subspace of the plane.
- Step 1
She checks the first condition, which asks whether the origin is on the line.
Step 4: Your Turn
Practice makes it stick.
The Line
Problem 1 of 2
The line y = 3x. Is it a subspace of the plane? 1 yes, 0 no.
The Shifted Line
Problem 2 of 2
The line y = 3x + 5. Is it a subspace? 1 yes, 0 no.
Subspace or Not
1 of 8
The set containing only the zero vector. Is it a subspace? 1 yes, 0 no.
2 of 8
The whole plane. Is it a subspace of itself? 1 yes, 0 no.
3 of 8
A disc of radius 1 about the origin. Is it a subspace? 1 yes, 0 no.
4 of 8
How many subspaces of the plane have dimension 2?
5 of 8
A matrix whose columns span the plane. What is the dimension of its column space?
6 of 8
An invertible matrix. What is the dimension of its null space?
7 of 8
Sort each set by whether it is a subspace of the plane.
Tap something to move it.
- Empty
- Empty
8 of 8
Does every subspace contain the zero vector? 1 yes, 0 no.
Step 5: Quick Check
Show what you know.
Question 1 of 2
The line y = −4x. Is it a subspace? 1 yes, 0 no.
Question 2 of 2
What is the null space of A?
What You Learned
- 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.