math 54 midterm 2 (001) professor paulin do not turn over …

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:

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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:

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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:

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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:

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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