|TIME: 3 HOURS|
ANSWER ALL QUESTIONS WITHOUT USE OF A CALCULATOR. CREDIT WILL BE GIVEN FOR METHOD AND PRESENTATION OF SOLUTIONS.
Row reduce the augmented matrix,
||Use elementary row operations to determine the inverse of A.
||Factor A as a product LU, wherein L denotes a lower
triangular matrix with unit diagonal entries and U denotes an upper
determine whether or not the vector
belongs to the subspace, span
||Find a basis for the column space of the matrix,
be three non-zero vectors from
Prove that if the vectors are mutually orthogonal then
they are linearly independent.
be two bases of
||Determine the change of coordinates matrix, P, for the change
||Given the coordinates,
for a vector
relative to ,
Determine an orthogonal base for the subspace
be a linear mapping for which
(i) ||Is T a one-to-one mapping? ||
Justify your answers to these questions.
be the linear
transformation for which
calculate the matrix,
relative to base .
Find the eigenvalues and eigenvectors of the
constitute a linearly independent set of
eigenvectors of the symmetric matrix,
||Verify that the eigenvectors are mutually orthogonal.
||Construct an orthogonal matrix P with which A is
orthogonally diagonalizable as A = PDP-1
for some diagonal matrix D.
||Find (i) the matrix P-1,
and (ii) the matrix D.