Linear Algebra
What is Linear Algebra?
A collection of vectors that is closed under vector addition and scalar multiplication.
Key formula / rule: Matrix Multiplication
Key points
- Understand the concepts of vectors, vector spaces, and linear independence.
- Perform matrix operations including addition, subtraction, multiplication, and inversion.
- Calculate determinants and understand their geometric and algebraic significance.
- Solve systems of linear equations using methods like Gaussian elimination.
Common exam trap
Confusing row operations with column operations when solving systems.
Definitions
- Term
Vector Space
- Meaning
A collection of vectors that is closed under vector addition and scalar multiplication.
- Term
Linear Independence
- Meaning
A set of vectors is linearly independent if the only linear combination that equals the zero vector is the trivial one (all scalars are zero).
- Term
Basis
- Meaning
A set of linearly independent vectors that spans a vector space.
- Term
Dimension
- Meaning
The number of vectors in a basis for a vector space.
- Term
Matrix
- Meaning
A rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
- Term
Determinant
- Meaning
A scalar value that can be computed from the elements of a square matrix and encodes certain properties of the linear transformation described by the matrix.
- Term
Eigenvalue
- Meaning
A scalar λ such that there exists a non-zero vector v for which Av = λv.
- Term
Eigenvector
- Meaning
A non-zero vector v such that Av = λv for some scalar λ (the eigenvalue).
Learning objectives
Understand the concepts of vectors, vector spaces, and linear independence.
Perform matrix operations including addition, subtraction, multiplication, and inversion.
Calculate determinants and understand their geometric and algebraic significance.
Solve systems of linear equations using methods like Gaussian elimination.
Compute eigenvalues and eigenvectors of matrices.
Analyze the properties of linear transformations.
Formulae
- Name
Matrix Multiplication
- Note
Order of multiplication matters.
- Expression
If A is m x n and B is n x p, then C = AB is m x p, where Cij = sum(Aik * Bkj) for k=1 to n.
- Name
Determinant of a 2x2 Matrix
- Note
- Expression
For A = [[a, b], [c, d]], det(A) = ad - bc.
- Name
Adjoint of a Matrix
- Note
Used in finding the inverse.
- Expression
The transpose of the cofactor matrix.
- Name
Matrix Inverse (for invertible square matrices)
- Note
det(A) must be non-zero.
- Expression
A⁻¹ = (1/det(A)) * adj(A)
- Name
Eigenvalue Equation
- Note
Equivalent to (A - λI)v = 0.
- Expression
Av = λv, where A is the matrix, v is the eigenvector, and λ is the eigenvalue.
- Name
Characteristic Equation
- Note
Roots of this equation are the eigenvalues.
- Expression
det(A - λI) = 0
- Name
Rank of a Matrix
- Note
- Expression
The number of linearly independent rows or columns, or the number of non-zero rows in its Row Echelon Form.
Prerequisites
Basic Algebra
Complex Numbers (optional but helpful)
Understanding of functions
Common mistakes
Confusing row operations with column operations when solving systems.
Incorrectly calculating determinants, especially for larger matrices.
Assuming a system of equations has a solution when it is inconsistent.
Misapplying properties of eigenvalues and eigenvectors.
Errors in matrix multiplication order or dimension compatibility.
Keywords
Vector
Matrix
Determinant
Eigenvalue
Eigenvector
Linear Transformation
System of Equations
Rank
Vector Space
Linear Independence
Gaussian Elimination
Practice preview
Given a matrix A = [[2, -2], [-2, 5]], find the eigenvalues of A. Then, determine which of the following statements about its eigenvalues is true.…
hard
Given two matrices A = [[2, 3], [1, 4]] and B = [[-1, 0], [2, 5]], calculate the matrix 2A - B.…
easy
For a 3x3 matrix A, if its eigenvalues are 1, 2, and 3, what is the trace of the matrix A?…
easy