Vector Spaces, Bases, and Dimension
Core definitions, axioms, and proof techniques for abstract vector spaces, span, independence, and dimension.
Keyboard Shortcuts
💡 Pro tip: Use keyboard shortcuts for faster studying!
Study Smart Tips for Vector Spaces, Bases, and Dimension
Master these concepts using proven study techniques that actually work:
Active Recall
Test yourself before flipping each card to strengthen memory retention
Spaced Repetition
Review difficult cards more frequently than easy ones
Multiple Sessions
Break study time into shorter, focused sessions
Explain Aloud
Verbalize answers to reinforce understanding
Questions Covered in This Set
12 cards to master
What is a vector space over a field F?
A set V with addition (V×V→V) and scalar multiplication (F×V→V) satisfying 8 axioms: commutativity, associativity, zero vector, additive inverses, 1v=v, a(bv)=(ab)v, and the two distributive laws. No length, angle, coordinates, or vector-vector product.
What must you check to prove W ⊆ V is a subspace?
Only three things (the other axioms are inherited): 0 ∈ W, W closed under addition, W closed under scalar multiplication. Equivalently: W nonempty and au + bv ∈ W for all u,v ∈ W, a,b ∈ F.
Prove 0v = 0 (scalar zero times v equals the zero vector).
0v = (0+0)v = 0v + 0v by axiom 8. Add −(0v) to both sides: 0 = 0v. ∎ Pure axiom-chasing — no coordinates used.
Why is {(x,y) : x + y = 1} not a subspace of ℝ²?
It does not contain the origin (0+0 = 0 ≠ 1), so it fails the 0 ∈ W requirement. Also not closed under addition or scaling.
Why is {(x,y) : xy = 0} not a subspace?
It contains (1,0) and (0,1) but not their sum (1,1), since 1·1 ≠ 0 — it fails closure under addition even though it contains 0.
Define span, and what kind of set is it?
The span of v₁,…,v_k is the set of all linear combinations a₁v₁ + ⋯ + a_kv_k. It is always a subspace — in fact the smallest subspace containing all the vᵢ.
Define linear independence.
v₁,…,v_k are independent if a₁v₁ + ⋯ + a_kv_k = 0 forces a₁ = ⋯ = a_k = 0. Otherwise they are dependent, meaning some vector is redundant (expressible via the others).
Are 1, cos²x, sin²x independent in C(ℝ)?
No. 1·1 + (−1)cos²x + (−1)sin²x = 0, the zero function — a nontrivial dependence relation.
Show 1, x, eˣ are linearly independent in C(ℝ).
Suppose a + bx + ceˣ = 0 for all x. Differentiate twice: ceˣ = 0 ⇒ c = 0; then b = 0, then a = 0. Note the tool was calculus, not row reduction.
What is a basis, and what is its key equivalent characterization?
A basis is a list that is linearly independent and spans V. Equivalently, B = (v₁,…,v_n) is a basis iff every v ∈ V has a unique expression as a₁v₁ + ⋯ + a_nv_n.
Name several non-ℝⁿ examples of vector spaces.
P_n (polynomials of degree ≤ n), P (all polynomials, infinite-dimensional), M₂ₓ₃(ℝ), C[0,1], the solution set of y'' + y = 0, and the null space {x : Ax = 0}.
Why is the solution set of y'' + y = 0 a vector space?
If y₁ and y₂ solve it, so does ay₁ + by₂ — closure under linear combinations. This closure is exactly why linear ODEs are tractable.
