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.