Comps Syllabus for Linear Algebra
Download linear algebra comps syllabus
Basic definitions
- Vector space
- Subspace
- Span of a finite subset
- Linear independence of a finite set of vectors
- Basis and dimension
- Linear transformation
- Kernel or null space
- Image or range
- Inverse of a matrix or linear transformation
- Determinant
- Characteristic polynomial
- Eigenvalues and eigenvectors of a matrix
- Diagonalizability
Computational techniques
- Determine when a subset is a subspace
- Basic matrix manipulations
- Row operations on matrices
- Solving systems of linear equations
- Find the inverse of a matrix
- Find a basis of a given subspace
- Find the nullity, rank, and determinant of a matrix
- Find the null space N(T) and range R(T) of a linear transformation T
- Given bases of V and W, find the matrix of a linear transformation T : V to W
- Given a matrix
- Compute its characteristic polynomial
- Find its eigenvalues and eigenspaces
Basic results to know
- dim N(T) + dim R(T) = dim V
- nullity (A) + rank (A) = number of columns of A
- Criteria for a matrix inverse to exist
- Criteria for A to be diagonalizable
Write short proofs for problems involving subspaces, linear maps, linear independence, spanning sets, null spaces and ranges.