# Linear Algebra

Need help with my Linear Algebra problems

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Need help with my Linear Algebra problems

Math 2318 Examination 1

Name: Due Date: September 19, 2020 @5:00 pm

Examination Instructions

Electronic Submission Protocols

The completed exam should be uploaded – as a single PDF file – to the CANVAS Assignments forum. Please see the submission protocols sent to you via email. However, I summarize the document naming protocols below:

File naming protocols:

Please put your “last name”, “first name”, “exam number”, and“course” in the file name – separated by underscores (without the quotes or commas). For example, the filename for the submission for student John Doe should be: Doe John Exam1 2318 .

Quality of Scanned Work: In this case, 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.

Instructions Show all work and pertinent calculations (particularly when performing row reduc- tion). 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 and state your conclusion on each problem. 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 your first draft. Once again, show all of your work and justify your solutions fully. In addition, the simple operational rules for this examination 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 1.1, 1.2, 1.3, 1.4, 1.5, 1.7, 1.8, and 1.9.

Honesty Policy

HCC students are expected to adhere to the academic integrity policies set forth in the student handbook (See Student Handbook for full details.). In brief, on this quiz and all other submitted work in this course, students are expected to submit their own original work and solutions to problems, discussions, etc. Thus, for example, it would not be appropriate to submit - as your own - work copied from another person, other sources (e.g., Solutions Manual or Instructor’s Edition), or copying work and/or solutions from a website, web application, or app. In brief, your submissions on this and other coursework should be of your own making.

–Created/Revised by Mr. Sever 9/18/20 Page 1 of 8

Math 2318 Examination 1

Name: Due Date: September 19, 2020 @5:00 pm

1. (2 pts.) The augmented matrix for a linear system is

1 0 0 11 0 0 1 4 0 1 0 −1

.

Write the solution of the associated linear system.

2. (6 pts.) The augmented matrix for a linear system is

1 0 0 0 2 0 0 1 −3 1 0 0 0 0 0

.

Write the general solution of the associated linear system.

3. (6 pts.) Identify the matrix as either REF (reduced echelon form), EF (echelon form, but not re- duced echelon form), or NA (not in echelon form).

(a)

1 0 0 11 0 0 1 −4 0 1 0 1

(b)

1 −2 0 0 1 1 0 0 1

(c)

[

2 2 2 2

]

(d)

1 −2 0 0 0 0 0 1 −3 0 0 0 0 0 1 0 0 0 0 0

(e)

0 1 0 0 0 1 0 0 0

(f)

[

0 0 0 0

]

4. (2 pts.) Let u =

−3 4 5

, v =

1 2

−3

, and w =

−5 −7

2

. Find 3u − 2v + 6w.

5. (3 pts.) Given that

4 13

−2

= 5

0 1

−3

− 3

−2 0

−1

+ 2

−1 4 5

and A =

0 −2 −1 1 0 4

−3 −1 5

.

Find a solution to Ax =

4 13

−2

.

–Created/Revised by Mr. Sever 9/18/20 Page 2 of 8

Math 2318 Examination 1

Name: Due Date: September 19, 2020 @5:00 pm

6. (10 pts.) Apply the row reduction algorithm to transform A =

1 −2 2 1 2 −2 4 −3 −2 −5 −1 2 0 0 −1

3 −6 9 5 9

into reduced echelon form. (Show all work.)

–Created/Revised by Mr. Sever 9/18/20 Page 3 of 8

Math 2318 Examination 1

Name: Due Date: September 19, 2020 @5:00 pm

7. (8 pts.) Let u =

−3 4 5

, v =

1 2

−3

, and w =

−5 −7

2

. Determine whether {u, v, w} is linearly

independent. (Show all work.)

8. (8 pts.) Let A =

[

h 2 6h2 1 2

h 3h

]

and b be an arbitrary 2 × 1 vector.

Find h so that the linear system given by [A | b ] is consistent for any choice of b.

–Created/Revised by Mr. Sever 9/18/20 Page 4 of 8

Math 2318 Examination 1

Name: Due Date: September 19, 2020 @5:00 pm

9. (8 pts.) Let A =

[

1 − 5 2

−2 1

2 0 3

]

, u =

−5 −7

2

and w =

30 −6

5

.

Define T : R3 → R2 be defined by T (x) = Ax.

a) Find T (u)

b) Find T (w)

c) Determine whether T is one-to-one. (Explain answer, of course.)

10. (6 pts.) Given that u, v, y and w ∈ R6 with 3w = 2y + 3u − 2v. Discuss the linear independence of {u, v, y, w}; that is determine whether {u, v, y, w} is linear independent or not.

11. (7 pts.) Let T

x1

x2

x3

x4

=

x4 − 3x3 + 2x1 3x4 + 5x3 + 6x2

x1 + 5x3 − 7x2 + x4

. Find a matrix that implements the mapping; i.e., find

the standard matrix for T .

–Created/Revised by Mr. Sever 9/18/20 Page 5 of 8

Math 2318 Examination 1

Name: Due Date: September 19, 2020 @5:00 pm

12. (10 pts.) Determine whether the linear transformation in the previous problem is onto. (Of course, show all work.).

13. (4 pts.) Suppose that T : R4 → R5 is a linear transformation where

T (2u) =

2 0

−6 8

and T (3v) =

0 −9 21 15

. Find T (w) where w = u − v.

–Created/Revised by Mr. Sever 9/18/20 Page 6 of 8

Math 2318 Examination 1

Name: Due Date: September 19, 2020 @5:00 pm

14. (20 pts.) True (T) or False (F). Circle the best response.

(a) T F An inconsistent system has more than one solution.

(b) T F Two linear systems are equivalent if they have the same number of free variables.

(c) T F The linear system associated with Ax = 0 is always consistent.

(d) T F A linear system of five equations in three unknowns cannot be consistent.

(e) T F Two matrices are row equivalent if they same number of rows.

(f) T F Every elementary row operation is reversible.

(g) T F The reduced echelon form of a matrix is unique.

(h) T F If u and v are in R4 where u is not a scalar multiple of v, then {u, v} is linearly independent.

(i) T F If u, v and w are in R4 where {u, v, w} is linearly independent, then {u, v} is linearly independent.

(j) T F If u, v and w are in R4 where {u, v} is linearly independent, then {u, v, w} is linearly independent.

(k) T F If A = [a1 a2 a3 a4] with b = −2a1 + 5a4, then Ax = b is consistent.

(l) T F If A = [a1 a2 a3 a4] with b = −2a1 + 5a4, then b is not in the span of the columns of A.

(m) T F If Ax = 0 has the trivial solution, then the columns of A are linearly independent.

(m) T F If −4y is a solution to Ax = −2b, then 2y is a solution to Ax = b.

(n) T F If Ax = b is inconsistent for some b, then A has a pivot position in each row.

(o) T F If T : R4 → R5 is a linear transformation, then T (0) = 0.

(p) T F If T is a linear transformation with T (w) = T (y) where w 6= y, then T is one-to-one.

(q) T F If T : R7 → R6 with T (x) = Ax and T (u) = 4v, then 4v is in the span of columns of A.

(r) T F If T : R2 → R3 with T (x) = Ax and T (u) = 4v, then v is in the span of columns of A.

(s) T F If T : R4 → R5 is a linear transformation and T (w) = 0 where w 6= 0, then T is not one-to-one.

–Created/Revised by Mr. Sever 9/18/20 Page 7 of 8

Math 2318 Examination 1

Name: Due Date: September 19, 2020 @5:00 pm

(Bonus): Let T : R4 → R3 and S : R3 → R4 be linear transformations defined by:

T

x1

x2

x3

x4

=

x3 − 3x4 + 2x2

3x3 + 5x4 + 6x1

x2 + 5x4 − 7x1 + x3

and S

x1

x2

x3

=

x2 − 3x1 + 2x4

x2 + x1

3x1 + 5x4

2x2 − x1 + 4x4

.

1. Describe the linear transformation H defined by H = T ◦ S (i.e., H(x) = (T ◦ S)(x)). You are not being asked to show that H is a linear transformation; rather, just find H .

2. Determine whether H is a) one − to − one and b) onto.

–Created/Revised by Mr. Sever 9/18/20 Page 8 of 8