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Name: Student Number:
(First) (Last), (CAPITAL letters please)
MATH 232
FINAL EXAMINATION
Date: December 8, 2004 Instructors: J. Lester and G. Sabin
Time: 8:30 a.m. - 11:30 a.m.
INSTRUCTIONS
Do not lift the cover page until instructed!
No aids or materials (including calculators or scrap paper) are allowed.
For full credit show intermediate steps in your answers.
All answers to be written directly on the question paper. Clearly strike
out any work you do not want marked. If you run out of space use the
back of the preceding page, but make sure you indicate that you have
done so on the front page. Otherwise, any work on the back page will be
considered rough work not part of your answer and will not be marked.
The exam consists of ten questions and 16 pages (counting this one). The
last page consists of the list of the vector space axioms.
Question Mark
1 /
Total /
MARKS
- Let S be the plane with the equation, (6 pts)
2 x + y z = 6
Find the shortest distance from the point P (3; 1 ; 1) to the plane S and the point
Q on the plane S closest to P:
(c) Find all values of k for which the following matrix is invertible, (2 pts)
B =
k
2 1 0
Note it is not required to determine the inverse.
(d) Let E be an n n matrix. Give Öve statements that are equivalent to the
statement (5 pts)
E is invertible.
- Let
A =
; H =
It is given that H and A are row-equivalent.
(a) What is the rank of A? (1 pt)
(b) Write down a basis for the row space of A (row(A)): (2 pts)
(c) Write down a basis for column space for A (col(A)) : (2 pts)
(d) Find a basis for the null space of A (null(A)). (4 pts)
(e) What is the nullity of A? (1 pt)
(f) Find a basis for the othogonal complement to row(A). (2 pts)
(a) Find a basis for the subspace of R
3 given by the plane, 3 x 2 y + 5z = 0. (3 pts)
(b) Let fu; vg be a basis for a vector space V. Explain why or why not each of
the following sets of vectors is a basis for V. (3 pts)
i. f 0 ; u + vg
ii. fu + v; u 2 vg
iii. fu; u + v; u vg
(c) Show that V = f(x; y) in R
2 jx 0 ; y 0 g with the usual vector addition and
scalar multiplication in R
2 ; is not a vector space. (3 pts)
(d) Find another vector that can be added to fv 1 ; v 2 g to form a basis in R
3 where (3 pts)
v 1 =
and v 2 =
(e) Is (3 pts)
W =
a b
a; b R
a subspace of the vector space of all 2 2 matrices? Justify your answer.
(a) Let T : R
2 ! R
3 be the a linear transformation deÖned by (3 pts)
T
5 ; T
Find T
(b) If D is the set of all real-valued di§erentiable functions show that T : D ! D
deÖned by T (f ) = f
0
- f is a linear transformation. (3 pts)
(c) Let V and W be vector spaces of dimension n: Prove that a one-to-one linear
transformation S : V ! W maps a basis for V to a basis for W. (4 pts)
- Let T : P 2 ! R
3 be a linear transformation deÖned by
T
a + bx + cx
2
2 a b
a + b
a
(a) Find a basis for range(T ): (2 pts)
(b) Find a basis for ker(T ): (2 pts)
(c) What is the rank of T? (1 pt)
(d) State the deÖnition of an isomorphism. (1 pt)
(e) Is T an isomorphism? Justify you answer. (2 pts)
iii. Find a matrix P that diagonalizes A and the resulting diagonal matrix,
D. Write an equation that relates A and D: (4 pts)
iv. Is A orthogonally diagonalizable? Why or why not? (2 pts)
(b) Prove that eigenvalues of similar matrices are the same. (4 pts)
(a) The following vectors form a basis for a subspace W of R
3 : (3 pts)
v 1 =
(^5) ; v 2 =
Do v 1 and v 2 form an orthogonal basis for W? Justify your answer. If v 1
and v 2 do not form an orthogonal basis, use one of these vectors and another
vector from W to form an orthogonal basis.
(b) Find an orthonormal basis for W in (a): (1 pt)
Vector Space Rules
A set of vectors, V; with operations of addition and scalar multiplication deÖned on
it, is a vector space if, for any vectors u; v and w in V and any scalar c; the following
rules hold:
- u + v is a vector in V
- cu is a vector in V
- u + v = v + u
- u + (v + w) = (u + v) + w
- There is a vector 0 in V such that u + 0 = 0 + u = u for all vectors u in V.
- For each vector u in V , there is a vector u in V such that u + ( u) = 0.
- c(u + v) = cu + cv
- (c + d)u = cu + du
- (cd)u = c(du)
- 1 u = u
