Linear Algebra: Vector Spaces and Matrices

Study Notes

Vector Spaces

Definition: A vector space is a collection of vectors that can be scaled and added together while satisfying specific axioms.
Axioms: Includes closure under addition and scalar multiplication, existence of zero vector, existence of additive inverses, and distributive properties.
Subspaces: A subset of a vector space that is also a vector space under the same operations.
Basis: A set of vectors in a vector space that is linearly independent and spans the space.
Dimension: The number of vectors in a basis; it indicates the size of the vector space.

Matrix Operations

Addition: Matrices of the same dimensions can be added by adding corresponding elements.
Scalar Multiplication: Each element of a matrix is multiplied by a scalar.
Matrix Multiplication: The dot product of rows and columns; requires the number of columns in the first matrix to equal the number of rows in the second.
Transpose: Flipping a matrix over its diagonal (rows become columns).
Inverse: A matrix A has an inverse A⁻¹ if A * A⁻¹ = I (identity matrix); exists only for square matrices and when determinant is non-zero.

Determinants

Definition: A scalar value that can be computed from the elements of a square matrix, providing insight into the matrix properties.
Geometric Interpretation: Represents the volume scaling factor for linear transformations; if det = 0, the matrix is singular, indicating no unique solution.
Calculation: For 2x2 matrices: det(A) = ad - bc; for 3x3 matrices, use the rule of Sarrus or cofactor expansion.
Properties:
- det(AB) = det(A) * det(B)
- det(A⁻¹) = 1/det(A)
- det(A^T) = det(A)

Eigenvalues And Eigenvectors

Eigenvectors: Non-zero vectors that only change by a scalar factor when a linear transformation is applied (Ax = λx).
Eigenvalues: Scalars (λ) associated with eigenvectors that represent the factor by which the eigenvector is scaled.
Characteristic Equation: To find eigenvalues, solve the equation det(A - λI) = 0.
Multiplicities: Algebraic multiplicity (number of times an eigenvalue appears) and geometric multiplicity (number of linearly independent eigenvectors associated with an eigenvalue).

Linear Transformations

Definition: A function T: V → W between two vector spaces that preserves vector addition and scalar multiplication.
Matrix Representation: Each linear transformation can be represented by a matrix, translating it to matrix operations.
Properties:
- T(u + v) = T(u) + T(v)
- T(cu) = cT(u) for a scalar c.
Kernel and Image:
- Kernel: The set of all vectors that map to the zero vector (null space).
- Image: The set of all vectors in the codomain that can be expressed as T(v) for v in the domain.
Invertibility: A linear transformation is invertible if its matrix representation is invertible.

Vector Spaces

A collection of vectors that can be scaled and added together while satisfying specific axioms
- Axioms include closure under addition and scalar multiplication, existence of zero vector, existence of additive inverses, and distributive properties
- Subspaces are subsets of a vector space that are also vector spaces under the same operations
- Basis is a set of linearly independent vectors in a vector space that spans the space
- Dimension is the number of vectors in a basis, representing the size of the vector space

Matrix Operations

Addition: Adding matrices of the same dimensions by adding corresponding elements
Scalar Multiplication: Multiplying each element of a matrix by a scalar
Matrix Multiplication: Dot product of rows and columns, requiring the number of columns in the first matrix to equal the number of rows in the second
Transpose: Flipping a matrix over its diagonal, making rows into columns
Inverse: Matrix A has inverse A⁻¹ if A * A⁻¹ = I (identity matrix), existing only for square matrices with non-zero determinant

Determinants

A scalar value calculated from the elements of a square matrix, providing insights into its properties
- Geometric Interpretation: Represents the volume scaling factor for linear transformations; if det = 0, the matrix is singular, indicating no unique solution
- Calculation: For 2x2 matrices: det(A) = ad - bc; for 3x3 matrices, use the rule of Sarrus or cofactor expansion
- Properties:
  - det(AB) = det(A) * det(B)
  - det(A⁻¹) = 1/det(A)
  - det(A^T) = det(A)

Eigenvalues And Eigenvectors

Eigenvectors: Non-zero vectors experiencing only a scalar factor change under a linear transformation (Ax = λx)
Eigenvalues: Scalars (λ) associated with eigenvectors, representing the scaling factor
Characteristic Equation: Solving det(A - λI) = 0 to find eigenvalues
Multiplicities:
- Algebraic Multiplicity: Number of times an eigenvalue appears
- Geometric Multiplicity: Number of linearly independent eigenvectors associated with an eigenvalue

Linear Transformations

A function T: V → W between two vector spaces preserving vector addition and scalar multiplication
Matrix Representation: Can be represented by a matrix which allows translation to matrix operations
Properties:
- T(u + v) = T(u) + T(v)
- T(cu) = cT(u) for a scalar c
Kernel and Image:
- Kernel: Set of vectors mapping to the zero vector (null space)
- Image: Set of vectors in the codomain representable as T(v) for v in the domain
Invertibility: A linear transformation is invertible if its matrix representation is invertible

Description

This quiz covers the fundamental concepts of vector spaces and matrix operations, including definitions, axioms, and properties. Explore subspaces, basis, dimension, and matrix operations such as addition and multiplication.