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📘 Lesson 6 of 7 · Linear Algebra

🌐 Vector Spaces

A vector space is a collection of objects (vectors) that can be added together and scaled, following a consistent set of rules — an abstraction that unifies everything from arrows in space to functions and polynomials.

Course progress: 86%

01 Key Concepts

Vector Space

A set of vectors that is closed under addition and scalar multiplication, satisfying rules like associativity, commutativity, and the existence of a zero vector.

Subspace

A subset of a vector space that is itself a vector space — it must contain the zero vector and be closed under addition and scalar multiplication.

Linear Combination

Any expression formed by scaling and adding vectors together, like 3*v1 + 2*v2 - v3.

Span

The set of every possible linear combination of a group of vectors — essentially everything you can 'reach' using just those vectors.

Linear Independence

A set of vectors is linearly independent if no vector in the set can be written as a combination of the others (the only way to get the zero vector is if every coefficient is zero).

Basis and Dimension

A basis is a linearly independent set of vectors that spans the entire space. The dimension is the number of vectors in a basis.

02 Solved Examples

Example 1 Is the set {(1,0), (0,1)} linearly independent in 2D space?
  1. Check if one vector is a scalar multiple of the other.
  2. (1,0) and (0,1) point in completely different directions; neither is a multiple of the other.
Answer: Yes, they are linearly independent
Example 2 Is the set {(1,2), (2,4)} linearly independent?
  1. Check if one vector is a scalar multiple of the other.
  2. (2,4) = 2*(1,2), so the second vector is a multiple of the first.
Answer: No, they are linearly dependent
Example 3 What is the dimension of standard 2D space (the plane)?
  1. A basis for 2D space needs exactly 2 linearly independent vectors, such as (1,0) and (0,1).
  2. The dimension equals the number of vectors in a basis.
Answer: Dimension = 2

03 Practice Questions

1Is {(1,0,0),(0,1,0),(0,0,1)} linearly independent?
Yes
2Is {(1,1),(2,2)} linearly independent?
No, the second is a multiple of the first
3What is the dimension of standard 3D space?
3
4What must a subspace always contain?
The zero vector
5What is a 'linear combination' of vectors?
A sum of scaled vectors, like a*v1 + b*v2

📄 Vector Spaces — Downloadable Worksheet

10 questions with a full answer key. Grab the PDF to print, or try the interactive version in your browser.

🗒 Cheat Sheet ⬇ Download PDF 🖥 Interactive Worksheet