Linear Algebra |
Test #2
|
May 5, 2003
|
Name____________________ |
R. Hammack
|
Score ______
|
(1) This problem concerns the matrix .
(a) Is
in the null space of A?
(b) Is [2 9 7 2] in the row
space of A?
(2)
(3) Find the inverse of this matrix without doing any row reductions.
(Hint: is it orthogonal?)
(4) Suppose . Find
.
(5) Consider the matrix
(a) Find the eigenvalues of A.
(b) Find the eigenspaces of A.
(c) Diagonalize the matrix A, that is find an invertible matrix
P and a diagonal matrix D with .
(6) Suppose is
an orthogonal basis for
having
the property that
,
,
and
.
Suppose also that
satisfies
,
,
and
.
Find
.
(7) Suppose is
a basis for
, and
is a linear transformation for which
,
,
and
.
Moreover, suppose v is a vector in
for
which
. Find
T(v).
(8) Suppose A is a matrix which has 5 rows, and the rows are linearly
independent. Suppose also that Null(A) is two-dimensional. How many columns
does A have?
(9) Suppose A is a 5×5 matrix with a one-dimensional null
space. Find det(A).