Concepts: State the definitions and use them in proofs.

• minor, cofactor, determinant
• classical adjoint of a matrix
• eigenvalue, eigenvector, eigenspace
• characteristic polynomial
• algebraic and geometric multiplicity of an eigenvalue
• diagonalizable matrix
• vector space, vector subspace
• linear combination of vectors
• span of a set of vectors
• linearly dependent set, linearly independent set

Algorithms: Do the following.

• Evaluate the determinant of a matrix using
  • Laplace expansion along any row or column;
  • the shortcut for triangular matrices;
  • row reduction for general matrices;
• Use determinants to find: the signed area of a parallelogram; the signed volume of a parallelopiped.
• Use the relation between |A| and |A^T|.
• Use the relation between |A| and |A^(-1)| when A is nonsingular.
• Use the adjoint matrix to find the inverse of a matrix.
• Use Cramer's Rule to represent the solution of a nonsingular linear system by determinants.
• Determine whether a (real or complex) matrix is diagonalizable; if it is, transform it to diagonal form.
• Use row reduction to determine whether a vector lies in the span of a set of vectors.
• Use row reduction to determine whether a set of vectors is linearly dependent, and to find a nontrivial linear relation if possible.

Theorems: State the theorems and use them in proofs.

• Nonsingular Matrix Theorem: relate nonsingularity of matrix A to
  • Rank(A)
  • |A|
  • Row equivalence of A to I
  • Number of solutions of AX=O
  • Unique solvability of AX=B for all B in R^n.
  • Linear independence of the rows of A.
  • Linear independence of the columns of A.
• Diagonalization Theorem (3.15).

alex at iastate dot edu

Document last modified on Mon Mar 3 2008