Math 254

  • Solve systems of linear equations using several algebraic methods.
  • Construct and apply special matrices, such as symmetric, skew-symmetric, diagonal, upper triangular or lower triangular matrices.
  • Perform a variety of algebraic matrix operations, including multiplication of matrices, transposes, and traces.
  • Calculate the inverse of a matrix using various methods, and perform application problems involving the inverse.
  • Compute the determinant of square matrices and use the determinant to determine invertibility.
  • Derive and apply algebraic properties of determinants.
  • Perform vector operations on vectors from Euclidean Vector Spaces including vectors from R^n.
  • Compute the equations of lines and planes and write these in their corresponding vector forms.
  • Perform linear transformations in Euclidean vector spaces, including basic linear operatons, and determine the standard matrix of the linear transformation.
  • Prove whether a given structure is a vector space and determine whether a given subset of a vector space is itself a vector space.
  • Determine if a set of vectors spans a space, and if such a set is linearly dependent or independent.
  • Determine if a set of functions is linearly independent using various techniques including calculating the determinant of the Wronskian.
  • Solve for the basis and the dimension of a vector space.
  • Determine the rank, the nullity, the column space and the row space of a matrix.
  • Describe orthogonality between vectors in an abstract vecotr space by means of an inner product, and compute the inner product between vectors of this inner product space.
  • Compute the QR-decomposition of a matrix using the Gram-Schmidt process.
  • Perform changes of bases for a vector space, including computation of the transition matrix and determining an orthonormal basis for the space.
  • Compute all the eigenvalues of a square matrix, including any complex eigenvalues and determine their corresponding eigenvectors..
  • Determine if a square matrix is diagonalizable and compute the diagonalization of a matrix whose eigenvalues are easily calculated.
  • Perform linear transformations among abstract general vector spaces, determining the rank, the nullity and the associated matrix of the transformation.