Linear Algebra
What is Linear Algebra?
A set V with two operations, vector addition and scalar multiplication, satisfying ten axioms.
Key points
- Understand the definition and axioms of a vector space.
- Identify common examples of vector spaces.
- Define and identify subspaces.
- Grasp the concepts of linear independence, span, basis, and dimension.
Common exam trap
Forgetting to check all ten axioms when verifying if a set is a vector space.
Definitions
- Term
Vector Space
- Meaning
A set V with two operations, vector addition and scalar multiplication, satisfying ten axioms.
- Term
Subspace
- Meaning
A non-empty subset W of a vector space V that is itself a vector space under the same operations.
- Term
Linear Independence
- Meaning
A set of vectors {v1, v2, ..., vk} is linearly independent if the only solution to c1*v1 + c2*v2 + ... + ck*vk = 0 is c1 = c2 = ... = ck = 0.
- Term
Span
- Meaning
The span of a set of vectors S is the set of all possible linear combinations of vectors in S.
- Term
Basis
- Meaning
A set of vectors that is linearly independent and spans the vector space.
- Term
Dimension
- Meaning
The number of vectors in any basis of a vector space.
Learning objectives
Understand the definition and axioms of a vector space.
Identify common examples of vector spaces.
Define and identify subspaces.
Grasp the concepts of linear independence, span, basis, and dimension.
Prerequisites
Basic set theory.
Properties of real and complex numbers.
Familiarity with basic algebraic operations.
Common mistakes
Forgetting to check all ten axioms when verifying if a set is a vector space.
Assuming a subset is a subspace without verifying closure under addition and scalar multiplication.
Confusing linear independence with the existence of a basis.
Incorrectly calculating the dimension of a vector space.
Keywords
Vector Space
Subspace
Linear Independence
Span
Basis
Dimension
Axioms
Scalar Multiplication
Vector Addition
Field
Practice preview
The determinant of the matrix [[2, 3], [4, 5]] is:…
easy
If A is a square matrix such that A = A^T, where A^T denotes the transpose of A, then A is called a:…
easy
Given two matrices A = [[1, 2], [3, 4]] and B = [[5, 6], [7, 8]], find the sum A + B.…
easy