Math 355 Final Review Sheet







Solutions of systems of linear equations
- What techniques do we have to find these solutions, when they exist (existence)?
- When are they unique (uniqueness)?
- Homogeneous vs. non-homogeneous
- Parametric vector form of a solution
Different representations of systems of linear equations
- Vector equation
- Matrix equation
Matrices
- Properties of matrix addition, scalar multiplication
- Matrix multiplication: different representations (e.g. row-column multiplication
vs. writing columns)
- Properties of matrix multiplication
- Transpose of a matrix, properties
- Inverse of a matrix
- How can you tell if a matrix is invertible?
- The Invertible Matrix Theorem
- Determinant of a matrix: cofactor expansion, properties
- Cramer’s Rule
Vector spaces
- Definition
- Examples?
- Dimension of a vector space
Subspaces
- Definition of a subspace
- Span{v1, …, vn), linear combinations
- Column space, Null space: differences between these (p. 232)
- Rank of a matrix
- The Rank Theorem
Linear dependence and linear independence
- Definitions
- Useful theorems?
- Basis
- The Spanning Set Theorem, The Basis Theorem
Linear transformations
- Definitions
- Domain, range, codomain, kernel
- Matrix transformation
- When is a transformation linear?
- Matrix of a linear transformation (standard matrix)
- One-to-one and onto: How did we prove these??
- Matrix for a linear transformation relative to bases B and C







Coordinate Systems
- Coordinate vector of x relative to a basis B
- The Unique Representation Theorem
- Change of coordinates matrix, change of basis
- Coordinate mapping
Eigenvalues and eigenvectors
- Definitions
- Eigenspace
- Characteristic polynomial, characteristic equation, multiplicity of an eigenvalue
- Finding eigenvalues and eigenvectors
- Complex eigenvalues
Diagonalization
- Similar matrices
- The Diagonalization Theorem
- Steps to diagonalize a matrix (if possible)
- Diagonal Matrix Representation Theorem
Diagonalization of Symmetric Matrices
- Orthogonally diagonalizable
- The Spectral Theorem
Inner product spaces
- Dot product in Rn, properties
- General inner product, properties
- Length/norm, unit vectors
- Normalizing
- Distance between vectors
- Orthogonal vectors, orthogonal sets & bases, orthonormal sets & bases
- Orthogonal complement of a subspace, W
- Orthonormal columns of a matrix
- Properties of orthogonal sets, Theorem 5 (p. 385)
- Orthogonal projection
- Orthogonal Decomposition Theorem, Best Approximation Theorem
- Gram-Schmidt process
- Cauchy-Schwarz Inequality, Triangle Inequality
Quadratic Forms
- Definition, matrix of a quadratic form
- Change of variable
- The Principal Axes Theorem
- Positive definite, negative definite, indefinite
Jordan Canonical Form
- “Almost diagonal”, what does the matrix look like?
- For the right basis as the columns of P, A = PJP-1
- Generalized eigenvectors, generalized eigenspaces
- Cycle of generalized eigenvectors
-Similar matrices have the same Jordan canonical form
Proof techniques we’ve talked about:
 For an “If-Then” statement, assume the “If”, prove the “Then”
 Proof by contrapositive
 Proof by contradicton
 Proof by induction
 Proof of if and only if statements: use a circle (or a bunch of little circles!)
Other notes about proofs:
 Always use all of your assumptions
 If you’re stuck, start with the definitions; often times the proofs in the book are
the “slickest” way to do things, but maybe not the only way
 Make sure you are explicit about the logic and reasoning in your proofs