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MATH 54 MIDTERM 2 (001)
PROFESSOR PAULIN
DO NOT TURN OVER UNTILINSTRUCTED TO DO SO.
CALCULATORS ARE NOT PERMITTED
YOU MAY USE YOUR OWN BLANKPAPER FOR ROUGH WORK
SO AS NOT TO DISTURB OTHERSTUDENTS, EVERYONE MUST STAYUNTIL THE EXAM IS COMPLETE
REMEMBER THIS EXAM IS GRADED BYA HUMAN BEING. WRITE YOUR
SOLUTIONS NEATLY ANDCOHERENTLY, OR THEY RISK NOT
RECEIVING FULL CREDIT
THIS EXAM WILL BE ELECTRONICALLYSCANNED. MAKE SURE YOU WRITE ALLSOLUTIONS IN THE SPACES PROVIDED.YOU MAY WRITE SOLUTIONS ON THEBLANK PAGE AT THE BACK BUT BESURE TO CLEARLY LABEL THEM
Name and Student ID:
GSI’s name:
Math 54 Midterm 2 (001)
This exam consists of 5 questions. Answer the questions in thespaces provided.
1. (25 points) (a) Let A =
⎛
⎜⎜⎝
0 1 −1 1 1 30 −1 1 0 −2 −10 0 0 2 −2 60 3 −3 3 3 8
⎞
⎟⎟⎠. Calculate bases for Ker(TA) and
Range(TA).
Solution:
(b) What is Rank(TA)? What is Nullity(TA)?
Solution:
PLEASE TURN OVER
Math 54 Midterm 2 (001), Page 2 of 5
2. (25 points) Let
B = {1, 1− x, 1 + x− x2, x3 − x} ⊂ P3(R), C = {x2 − x, x+ 2,−2} ⊂ P2(R)
be bases respectively. Let T be the following linear transformation:
T : P3(R) → P2(R)p(x) $→ xp′′(x)− p′(x)
Determine the matrix of T with respect to bases B and C. Is T onto? Justify youranswer.
Solution:
PLEASE TURN OVER
Math 54 Midterm 2 (001), Page 3 of 5
3. Let A =
⎛
⎜⎜⎝
1 −1 2 10 2 a 10 0 1 b0 0 0 2
⎞
⎟⎟⎠. For what values of a and b is A diagonalizable? You do not
need to diagonalize A. Justify your answer.
Solution:
PLEASE TURN OVER
Math 54 Midterm 2 (001), Page 4 of 5
4. Let A =
⎛
⎝−1 −1 00 1 1−1 0 1
⎞
⎠ and C = {
⎛
⎝101
⎞
⎠ ,
⎛
⎝−110
⎞
⎠ ,
⎛
⎝010
⎞
⎠}. Find a basis B such that
AB,C =
⎛
⎝0 1 01 1 00 0 0
⎞
⎠ .
Solution:
PLEASE TURN OVER
Math 54 Midterm 2 (001), Page 5 of 5
5. Let W be the span of the vectors
⎛
⎜⎜⎝
120−3
⎞
⎟⎟⎠ ,
⎛
⎜⎜⎝
0−211
⎞
⎟⎟⎠ ,
⎛
⎜⎜⎝
013−4
⎞
⎟⎟⎠ in R4. Calculate the minimum
distance between
⎛
⎜⎜⎝
3010
⎞
⎟⎟⎠ and W? Hint: This problem can be solved without applying
Gram-Schmidt.
Solution:
END OF EXAM
