Topic 3 of 8
Vector spaces, basis, and dimension
Step 19 · Linear algebra topics
Decide which vectors can be built from a set, whether a set has vectors to spare, and find a basis and the dimension of a subspace.
Span and linear independence
A linear combination of vectors adds up multiples of them: c₁v₁ + c₂v₂ + … The span of a set is everything you can build this way. The set is linearly independent if none of its vectors is a combination of the others.
Example: In the span?
Is ⟨7, 5⟩ in the span of ⟨1, 1⟩ and ⟨2, 1⟩?
- Solve a⟨1, 1⟩ + b⟨2, 1⟩ = ⟨7, 5⟩: a + 2b = 7 and a + b = 5.
- Subtracting gives b = 2, and then a = 3.
AnswerYes: 3⟨1, 1⟩ + 2⟨2, 1⟩
Example: Dependent
Are ⟨1, 2, 3⟩, ⟨2, 4, 6⟩ and ⟨0, 1, 1⟩ independent?
- ⟨2, 4, 6⟩ is 2 times ⟨1, 2, 3⟩.
- So one vector is a combination of the others.
AnswerNo, they are dependent
Subspaces
A subspace is a set of vectors that contains 0 and is closed under adding and scaling: combinations of its vectors stay inside it. In ℝ³ the subspaces are the origin alone, lines and planes through the origin, and ℝ³ itself.
Example: A plane
Is the set of ⟨x, y, z⟩ with x + y + z = 0 a subspace?
- It contains ⟨0, 0, 0⟩.
- If two vectors have coordinates adding to 0, so do their sum and any multiple of either.
AnswerYes: a plane through the origin
Example: Not a subspace
Is the set of ⟨x, y⟩ with y = x + 1 a subspace?
- It does not contain ⟨0, 0⟩, since 0 is not 0 + 1.
AnswerNo: the line misses the origin
Basis and dimension
A basis of a subspace is an independent set that spans it: enough vectors to build everything, with none to spare. Every basis of a given subspace has the same number of vectors, called its dimension.
Example: The standard basis
Give a basis of ℝ³ and its dimension.
- ⟨1, 0, 0⟩, ⟨0, 1, 0⟩ and ⟨0, 0, 1⟩ are independent and build any ⟨a, b, c⟩.
AnswerThose three vectors; dimension 3
Example: A basis for a plane
Find a basis for the plane x + y + z = 0.
- z = −x − y, so every vector in it is x⟨1, 0, −1⟩ + y⟨0, 1, −1⟩.
- These two vectors are independent.
Answer⟨1, 0, −1⟩ and ⟨0, 1, −1⟩; dimension 2
Example: Rank and nullity
A 3 × 5 matrix has 3 pivots. What are the dimensions of its column space and null space?
- The column space has one basis vector per pivot: 3.
- The null space has one per free variable: 5 − 3 = 2.
Answer3 and 2
Try it yourself
Work each one on paper first, then open the answer to check.
Is ⟨4, 6⟩ in the span of ⟨2, 3⟩?
Show answer
Yes: it is 2⟨2, 3⟩
Are ⟨1, 0⟩ and ⟨1, 1⟩ independent?
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Yes
Can 4 vectors in ℝ³ be independent?
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No: at most 3 can be
Is the set of ⟨x, y⟩ with xy = 0 a subspace?
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No: ⟨1, 0⟩ + ⟨0, 1⟩ is not in it
What is the dimension of the plane x − 2y + z = 0?
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2
A 4 × 6 matrix has 4 pivots. What is the dimension of its null space?
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2