Linear Algebra
What is Linear Algebra?
A non-empty set V over a field F, with vector addition (+) and scalar multiplication (·) operations satisfying ten specific axioms.
Key points
- Understand the definition and axioms of a vector space.
- Identify and provide examples of common vector spaces.
- Determine if a subset is a subspace.
- Test for linear independence and find the span of a set of vectors.
Common exam trap
Forgetting to check all ten axioms when verifying if a set is a vector space or subspace.
Definitions
- Term
Vector Space
- Meaning
A non-empty set V over a field F, with vector addition (+) and scalar multiplication (·) operations satisfying ten specific axioms.
- Term
Subspace
- Meaning
A subset W of a vector space V that is itself a vector space under the same operations of addition and scalar multiplication.
- Term
Linear Independence
- Meaning
A set of vectors {v1, v2, ..., vn} is linearly independent if the only solution to the equation c1v1 + c2v2 + ... + cnvn = 0 is c1 = c2 = ... = cn = 0.
- Term
Span
- Meaning
The span of a set of vectors S = {v1, v2, ..., vn} is the set of all possible linear combinations of these vectors.
- Term
Basis
- Meaning
A set of vectors that is linearly independent and spans the entire vector space.
- Term
Dimension
- Meaning
The number of vectors in any basis for a vector space.
Learning objectives
Understand the definition and axioms of a vector space.
Identify and provide examples of common vector spaces.
Determine if a subset is a subspace.
Test for linear independence and find the span of a set of vectors.
Find a basis and determine the dimension of a vector space.
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 or subspace.
Confusing the span of a set with the set itself.
Incorrectly determining linear independence/dependence.
Miscalculating the dimension of a vector space.
Assuming that any set of vectors that spans a space is a basis (it must also be linearly independent).
Keywords
Vector Space
Subspace
Linear Independence
Span
Basis
Dimension
Field
Axioms
Linear Combination
Practice preview
Find the determinant of the matrix A = [[1, 2], [3, 4]].…
easy
What is the rank of the matrix A = [[1, 0, 0], [0, 1, 0], [0, 0, 0]]?…
easy
What are the eigenvalues of the matrix A = [[1, 2], [3, 0]]?…
medium