MATH 2213

FINAL EXAMINATION December 1997 | TIME: 3 HOURS |

**SHOW ALL WORK. YOUR METHOD IS MORE IMPORTANT THAN THE RIGHT
ANSWER.**

MARKS | |||

1. | Let | ||

(2) | (a) | Find a basis for Nul A.
| |

(2) | (b) | Find a basis for Col A.
| |

(2) | (c) | Find a basis for Row A.
| |

(2) | (d) | What is the rank of A?
| |

Let , and . | |||

(3) | (a) | Prove that B is a basis for .
Quote any results you might use.
| |

(2) | (b) | Find . | |

(3) | (c) | Find a matrix P such that
and a matrix Q such that
for all
in .
(You may use part (c) to do part (b) if you wish.) | |

3. | Let | ||

(2) | (a) | Find elementary matrices
(k is the number of elementary
matrices that you need) such thatU is upper triangular.
| |

(2) | (b) | Find a lower triangular matrix L, such that A
= LU. (U as in part (a)).
| |

(4) | 4. | Why is the following set a subspace of ? Find a basis for it. | |

(4) | 5. | Let | |

Show that
is an eigenvalue for A and find a basis for
the corresponding eigenspace.
| |||

Orthogonally diagonalize the matrix What easily checked property of this matrix guarantees that it is orthogonally diagonalizable? | |||

(4) | 7. | Find an orthonormal basis for the subspace of spanned by | |

(4) | 8. | Let | |

Show that
is not in Col (A) and find a least squares solution
of
.
| |||

(4) | 9. | Let and be bases for a 2 dimensional subspace of . Suppose and . Find the change of coordinate matrices and . | |

(4) | 10. | Let T be a linear transformation from
onto .
If
are independent vectors in ,
prove that
are also independent vectors
in .
| |

(2) | 11. | Let A be a
matrix. If null(A)
has dimension 2, what are the dimensions of the row and column spaces
of A?
| |

(4) | 12. | Suppose is a solution to the system of equations (i.e. ). Prove that any solution to can be written where is a solution to . | |

13. | A is an m by n matrix of rank r. Suppose
has
no solution for some right sides
and infinitely many
solutions for some other right sides .
| ||

(2) | (a) | Decide whether the nullspace of A contains
only the zero vector and why.
| |

(2) | (b) | Decide whether the column space of A is all
of
and why.
| |

(2) | (c) | Can there be a right side for which has exactly one solution? Why or why not? | |

(60) |