Linear Algebra
What is Linear Algebra?
A non-empty set V over a field F, with binary operations of vector addition and scalar multiplication satisfying ten axioms.
Key points
- Define a vector space and list its axioms.
- Identify whether a given set with defined operations forms a vector space.
- Understand the concept of a subspace and verify if a subset is a subspace.
- Recognize common examples of vector spaces.
Common exam trap
Forgetting to check all ten axioms when verifying if a set is a vector space.
Definitions
- Term
Vector Space
- Meaning
A non-empty set V over a field F, with binary operations of vector addition and scalar multiplication satisfying ten axioms.
- Term
Field
- Meaning
A set with two operations (addition and multiplication) satisfying properties similar to those of real or complex numbers, allowing for division by non-zero elements.
- Term
Subspace
- Meaning
A subset of a vector space that is itself a vector space under the same operations of addition and scalar multiplication.
- Term
Zero Vector
- Meaning
The additive identity element in a vector space, denoted by 0, such that v + 0 = v for all vectors v.
Learning objectives
Define a vector space and list its axioms.
Identify whether a given set with defined operations forms a vector space.
Understand the concept of a subspace and verify if a subset is a subspace.
Recognize common examples of vector spaces.
Prerequisites
Basic set theory.
Properties of real and complex numbers.
Understanding of functions and 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 it contains the zero vector and is closed under addition and scalar multiplication.
Confusing the field of scalars with the set of vectors.
Incorrectly applying distributive or associative properties.
Keywords
Vector Space
Field
Scalar Multiplication
Vector Addition
Axioms
Subspace
Zero Vector
Linear Independence
Basis
Dimension
Practice preview
The sum of the eigenvalues of the matrix A = [[3, 2, 1], [0, 4, 5], [0, 0, 6]] is:…
easy
For what value of 'x' is the matrix A = [[2, x], [4, 8]] singular?…
easy
What is the maximum possible rank of a 3x4 matrix?…
easy