Linear Algebra · Lesson 8
Vector Spaces & Subspaces
A vector space is any set of objects where addition and scalar multiplication are defined and satisfy eight axioms. Understanding subspaces, span, linear independence, and basis reveals the geometric structure hidden inside every matrix.
Quick Check
1. Which of the following is always a subspace of ℝ²?
A line not passing through the origin
A line through the origin
A circle centered at the origin
Any single point other than the origin
2. Are (2, 4) and (1, 2) linearly independent?
Yes, they point in different directions
No, (2,4) = 2·(1,2) — they are parallel
Yes, because neither is the zero vector
It depends on the scalar field
3. The dimension of ℝ³ is:
2
3
1
9
4. The span of {(1,0), (0,1)} is:
A line
All of ℝ²
Only the two given vectors
ℝ³