Subspaces
A subspace is a smaller space inside a larger vector space.
For a set to be a subspace, it must satisfy three practical checks:
- it contains the zero vector
- adding two vectors in the set stays in the set
- scaling a vector in the set stays in the set
These rules keep the set stable under linear operations.
Simple Examples
In the plane, a line through the origin is a subspace.
The same line shifted away from the origin is not a subspace, because it does not contain zero.
In three dimensions, a plane through the origin can be a subspace.
The set of points on the line y = 2x contains [1, 2] and [3, 6].
Their sum is [4, 8]. Does the sum still lie on the same line?
Compute it first, then check your number.
HintCheck the rule
Check whether the second coordinate is still twice the first.
SolutionClosed under this addition
For [4, 8], the second coordinate is 2 * 4 = 8. Yes, it lies on the
same line. Enter 1.
Why The Origin Matters
A subspace must contain zero.
The line:
y = 2x
contains [0,0], so it passes this check.
The shifted line:
y = 2x + 1
does not contain [0,0], so it is not a subspace.
Enter 1 if the line y = 2x + 1 contains the zero vector [0,0], or 0
if it does not.
Compute it first, then check your number.
HintSubstitute zero
Check whether 0 = 2 * 0 + 1.
SolutionZero fails
For [0,0], the left side is 0, but the right side is 1. The zero
vector is not on the shifted line, so it is not a subspace.
Why Subspaces Matter
Subspaces appear when a model, matrix, or dataset only uses part of a larger space.
For example:
- a matrix may map many inputs into a lower-dimensional output region
- a dataset may vary mostly along a few directions
- a representation may contain a useful component and an irrelevant component
Subspaces help us name the part of a space where action actually happens.
The zero-vector rule is not arbitrary. If a set is closed under scaling, then
scaling any vector in the set by 0 should keep you inside the set. That result
is the zero vector. So any true subspace must include zero.
The vector [2,4] lies on the line y = 2x. If we scale it by 3, do we
stay on the same line? Enter 1 for yes, 0 for no.
Compute it first, then check your number.
HintScale then test
3[2,4] = [6,12].
SolutionClosed under scaling
First scale the vector:
Then check the line equation. Since 12 = 2 * 6, the scaled vector still
satisfies y = 2x, so it stays on the line.
Enter 1 if closure under scaling explains why every subspace must contain
the zero vector.
Compute it first, then check your number.
HintScale by zero
If v is in the subspace, 0v must also be in the subspace.
SolutionZero from scaling
Enter 1. Closure under scaling says that if is in the subspace, then
must be in the subspace. But is the zero vector.
Before Moving On
A subspace is not just a smaller region. It is a smaller region that behaves well under vector addition and scaling.