Linear Algebra and Theorems Quiz

10 QUESTIONS. linear algebra. Please solve it handwritten and scan it back to me. I need correct work and clear handwriting.

Name:
Linear Algebra
Due Date: 4/15/2020 @ 11:30 pm
Quiz 10 – Part 2, 40 Points
Electronic Submission Protocols
Please follow the following guidelines for submitting your quiz as a A SINGLE PDF FILE.
1. Student Name:Write your name (“First Name Last Name” or “Last Name, First Name”) on each page.
2. File Name: Please put your last name, first name, quiz number, submission number, and course in the file name –
separated by underscores. For example, the filename should be: Doe John Quiz10 P2 V1 2318.
The submission number should indicate which version is the latest submission.
3. Uploading your Submission: Before you send the email you must first examine the PDF file to make sure that your
work is legible and that there are no large black borders surrounding the images (this will waste too much toner when
printing). In other words, make sure that your work is legible in the scanned image and please try to ensure that I will not
be wasting printer toner when printing – Thank you.
Please upload your quiz to the Document Sharing area in MyMathLab (MML) into the relevant folder.
Instructions Show all work and pertinent calculations (particularly when performing elementary row operations).
An answer is not considered to be correct without complete and correct supporting work. In other words, for each question, to
receive full credit you must show all work. Explain your reasoning fully and clearly. I expect your solutions to be well-written,
neat, and organized. Do not turn in rough drafts. What you turn in should be the “polished” version of potentially several
drafts. Once again, show all of your work and justify your solutions fully. In addition, the simple operational rules for this
quiz are:
A. Calculators or computer software solutions will not be accepted. Students are expected to present their own analytical
solutions.
B. You may refer to theorems in the book (unless the question specifically states otherwise), but only cite theorems
that are from the appropriate sections in our textbook; i.e., those that come from sections 4.5, 4.6, and 4.7 or previous
sections.
Nota Bene:
To receive full credit on graded work, it is important that you write each step of the row reduction process correctly.
Furthermore, when completing a proof or justifying your work it is important that you explicitly state your conclusion and
reasons. When performing elementary row operations, please use a style that is similar to that outlined in the example.
Notations
1. For n ∈ N, In represents the n × n identity matrix defined by
In ≡ [e1 e2 . . . en ]
where ei ∈ Rn is given by [ei ]j ≡
(
1, if i = j
0, if i 6= j
.
2. The null space of A is written Null(A).
3. The dimension of a vector space V may be written as dim(V ).
4. If B and C are ordered bases of a finite-dimensional vector space V , then the change-of-coordinates
matrix from B to C may be written as P .
C←B
5. The vector space of polynomials Pn (t) is defined as
Pn (t) ≡ {an tn + an−1 tn−1 + · · · + a1 x + a0 : an , an−1 , . . . , a1 , a0 ∈ R}.
The author writes this space as Pn .
This portion (i.e., Part 2) of the entire quiz is worth 40 points.
Do not submit this cover page.
–Created/Revised by Mr. Sever 4/9/20
Page 0 of 5
Name:
Linear Algebra
Due Date: 4/15/2020 @ 11:30 pm
Quiz 10 – Part 2, 40 Points
1. (6 pts.) Let H =












2s − 3t
0
5s + 2t
s





: s, t ∈ R





.




(a) Show that H is a subsapce of R4 . Cite the theorems that you use to draw your conclusion or
justify your logic.
(b) State the dimension of H (i.e., state dim(H)).
–Created/Revised by Mr. Sever 4/9/20
Page 1 of 5
Linear Algebra
Quiz 10 – Part 2, 40 Points
Name:
Due Date: 4/15/2020 @ 11:30 pm
2. (2 pts.) Let V be a five-dimensional vector space, and let S be a subset of V which is
linearly independent. Complete the sentence by choosing exactly one of the following:
Then S
A) must have exactly five elements.
B) must consist of at least five elements.
C) must span V .
D) must have at most five elements.
E) must have infinitely many elements.
F) can have any number of elements (except zero).
G) must be a basis for V .
3. (2 pts.) Let V be a five-dimensional vector space, and let S be a subset of V which is
linearly dependent. Complete the sentence by choosing exactly one of the following:
Then S
A) must have exactly five elements.
B) must consist of at least five elements.
C) must span V .
D) must have at most five elements.
E) must have infinitely many elements.
F) can have any number of elements (except zero).
G) must be a basis for V .
H) None of A) through G) is correct.
4. (2 pts.) Let V be a three-dimensional vector space, and let S be a subset of V consisting of five
vectors. Complete the sentence by choosing exactly one of the following:
Then S
A) can span V , but only if S is linearly independent, and vice versa.
B) must be linearly dependent, and must span V .
C) must be linearly dependent, but may or may not span V .
D) may or may not be linearly independent, and may or may not span V .
E) cannot span V , but can be linearly independent or dependent.
F) must be linearly independent, but cannot span V .
G) must be linearly independent, but may or may not span V .
H) None of A) through G) is correct.
5. (1 pt.) Complete the sentence by choosing exactly one of the following options.
The rank of a 3 × 5 matrix
A) must be zero.
B) is three.
C) can be any number from zero to five.
D) can be any number from zero to two.
E) can be any number from two to five.
F) can be any number from zero to three.
G) must be two.
H) None of A) through G) is correct.
–Created/Revised by Mr. Sever 4/9/20
Page 2 of 5
Name:
Linear Algebra
Quiz 10 – Part 2, 40 Points
Due Date: 4/15/2020 @ 11:30 pm
6. (1 pt.) Complete the sentence by choosing exactly one of the following options.
The rank of a 5 × 3 matrix
A) can be any number from zero to two.
B) can be any number from zero to five.
C) must be zero.
D) can be any number from two to five.
E) is three.
F) can be any number from zero to three.
G) must be two.
H) None of A) through G) is correct.
7. (6 pts.) True or False
(a) T F
The columns of the change-of-coordinates matrix P are B-coordinate vectors in C.
C←B
2
(b) T F
If V = R and C = {c1 , c2 } and B = {b1 , b2 } are ordered basis for V , then row reduction
of [ c1 c2 b1 b2 ] to [ I2 P ] produces a matrix P that satisfies [x]B = P [x]C for all x ∈ V .
(c) T F
The dimension of P2 (t) is 2.
(d) T F
R2 is a two-dimensional subspace of R3
(e) T F
If A is a 4 × 7 matrix, then rank(A) + dim(Null(A)) = 7.
(f) T F
The rank of A is the dimension of the null space of A.
(g) T F
If H is a subspace of K, then dim(H) = dim(K).
(h) T F
If two matrices A and B are row equivalent, then their row spaces are equal.
(i) T F
The row space of A is the set of all linear combinations of the row vectors of A.
(j) T F
If there exists a set {v1 , v2 , . . . , vp } that spans V , then dim(V ) ≤ p.
(k) T F
If there exists a linearly independent {v1 , v2 , . . . , vp } in V , then dim(V ) ≤ p.
(l) T F
If every set of k elements in V fails to span V , then dim(V ) > k.
8. (3 pts.) Let B = {b1 , b2 } and C = {c1 , c2 } be ordered bases for a vector space V .
Suppose that b1 = 2c1 − 6c2 and b2 = 4c1 − 9c2 .
(a) Find the change-of-coordinates matrix from B to C, that is find P .
C←B
(b) Find [w]C for w = −3b1 + 2b2 . Use part (a).
–Created/Revised by Mr. Sever 4/9/20
Page 3 of 5
Name:
Linear Algebra
Due Date: 4/15/2020 @ 11:30 pm
Quiz 10 – Part 2, 40 Points
9. (6 pts.) Let B =
(”
5
7
# ”
,
−1
−3
#)
and C =
(”
−5
1
# ”
,
2
−2
#)
be ordered bases for a vector space R2 .
(a) Find the change-of-coordinates matrix from B to C, that is find P .
C←B
(b) Find the change-of-coordinates matrix from C to B, that is find P .
B←C
–Created/Revised by Mr. Sever 4/9/20
Page 4 of 5
Linear Algebra
Quiz 10 – Part 2, 40 Points
Name:
Due Date: 4/15/2020 @ 11:30 pm
10. (10 pts.) Let B = {1, 1 + t, 1 + t2 } and C = {2, 1 − t, 1 + 2t + 2t2 } be ordered bases for a vector
space P2 (t).
(a) Find the change-of-coordinates matrix from B to C, that is find P .
C←B
(b) Find the change-of-coordinates matrix from C to B, that is find P .
B←C
–Created/Revised by Mr. Sever 4/9/20
Page 5 of 5