MATH 54 MIDTERM 2 (PRACTICE 2)

PROFESSOR PAULIN

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Math 54 Midterm 2 (Practice 2)

This exam consists of 5 questions. Answer the questions in thespaces provided.

1. (25 points) (a) Let P3(R) be the vector space of polynomials of degree at most 3 withreal coefficients. Calculate the dimension of the subspace

U = Span(1 + x− x2, 2 + x2 + x3, 5− 2x− x3, 4− 3x+ x2 − x3}

Solution:

(b) Is U = P3(R)? Justify your answer.

Solution:

PLEASE TURN OVER

Math 54 Midterm 2 (Practice 2), Page 2 of 5

2. (25 points) (a) You are given a linear system with 5 equations in 6 unknowns. If thecorresponding homogeneous linear system has a solution set spanned by two linearlyindependent vectors, is it true that the original linear system is guaranteed to havea solution? If it is not possible, give an explicit example of such a system.

Solution:

(b) What about if we instead assume that the corresponding homogeneous linear systemsolution set is spanned by one non-zero vector? Justify your answer.

Solution:

PLEASE TURN OVER

Math 54 Midterm 2 (Practice 2), Page 3 of 5

3. (25 points) Let A =

⎛

⎜⎜⎝

1 0 0 20 1 1 00 0 1 01 0 0 1

⎞

⎟⎟⎠ and B = {

⎛

⎜⎜⎝

0−100

⎞

⎟⎟⎠ ,

⎛

⎜⎜⎝

0001

⎞

⎟⎟⎠ ,

⎛

⎜⎜⎝

1000

⎞

⎟⎟⎠ ,

⎛

⎜⎜⎝

0010

⎞

⎟⎟⎠}. Find a

basis C such that

AB,C =

⎛

⎜⎜⎝

0 0 0 −10 0 1 01 1 0 02 0 0 0

⎞

⎟⎟⎠ .

Solution:

PLEASE TURN OVER

Math 54 Midterm 2 (Practice 2), Page 4 of 5

4. (a) Give a precise statement of what it means for a square matrix A to be diagonalizable.

Solution:

(b) Is the matrix A =

⎛

⎜⎜⎝

1 2 0 00 3 0 00 0 2 10 0 0 2

⎞

⎟⎟⎠ diagonalizable? Justify your answer.

Solution:

PLEASE TURN OVER

Math 54 Midterm 2 (Practice 2), Page 5 of 5

5. Let W be the span of the vectors

⎛

⎜⎜⎝

1010

⎞

⎟⎟⎠ ,

⎛

⎜⎜⎝

001−1

⎞

⎟⎟⎠ in R4. Find an orthogonal basis for W⊥.

What is the minimum distance between

⎛

⎜⎜⎝

1110

⎞

⎟⎟⎠ and W?

Solution:

END OF EXAM