Math midterm 1 solution

(28 Points.) The following are True/False questions. For this problem only, you do not have to show any work. There will be no partial credit given for this problem. For this problem:

  • A correct answer gives 4 points.
  • An incorrect answer gives 0 points.
  • If you leave the space blank, you receive 2 points.

F (a) Five vectors in R5 always span R5.

F (b) If T : Rn → Rm and U : Rm → Rp are both linear transformations, then the standard matrix for the linear transformation U T is n × p.

T (c) If the Row Echelon form of a matrix A has a pivot in every column, then whenever A~x = ~b has a solution, it must be unique.

T (d) The sum of two solutions to the homogeneous problem A~x = ~0 is also a solution.

T (e) If a linear transformation T : Rn → Rm is both one-to-one and onto, then n must equal m.

F (f) Two vectors whose entries are all positive must be linearly independent.

T (g) AssumeT : Rn → Rm is a linear transformation and~v1,. . . ,~vp are vectors in Rn. If {T(~v1),. . . ,T(~vp)} are linearly independent, then {~v1,. . . ,~vp} are also linearly independent.

Each statement below is either true (in all cases) or false (in at least one case).

If false, construct a counterexample. If true, give a justification that carefully notes everywhere you use the definition of “linear (in)dependence.”

  • (12 Points.) If ~v1 and ~v2 are linearly independent and ~v1,~v2,~v3 are linearly dependent, then ~v3 is in Span ~v1,~v2 .
  • (12 Points.) If u~1 and u~2 are linearly dependent vectors in R2, then one of these vectors must be a constant multiple of the other.

(a) True. We want to show that ~v1 can be written as a linear combination of v1 and v2:

~v1,~v2,~v3 being linearly dependent means that there are numbers a1, a2, and a3 not all zero so that

a1~v1+ a2~v2+ a3~v3 = 0.

Note that a1 , 0 because otherwise we would have

a2~v2+ a3~v3 = 0,

where at least one of a2 or a3 is nonzero – which can’t happen because ~v1 and ~v2 are linearly independent.

So we’re safe to divide through by a1, giving:

a2 a3

~v1+ ~v2+ ~v3 = 0, a1 a1

and thus:

a2 a3

~v1 =− ~v2− ~v3; a1 a1

and so ~v1 is in Span ~v1,~v2 .

(b) True. Because u1 and u2 are linearly dependent, there are c1 and c2 not both zero so that

c1u~1+ c2u~2 = 0.

If c1 , 0, then

c2 u~1 =− u~2.


So u1 is a constant multiple of u2. If, on the other hand, c1 = 0, then c2 , 0, in which case:


u~2 =− u~1,


and thus u2 is a constant multiple of u1.

Consider the Linear Transformation T : R4 → R3 whose standard matrix is:

Math midterm 1 solution Image 1
  • (18 Points.) Is T onto?
  • (6 Points.) If T is onto, solve T(~x) = ~y for a nonzero vector ~y of your choice. If it is not onto, find a vector ~y for which ~y is not in the Range of T.

Reminder: you must show work to receive credit.

Let’s solve this entire problem in one computation (you don’t have to, but it’s more efficient to do so). In particular, we solve, for any ~b:

Math midterm 1 solution Image 2

(24 Points.) Consider the following system of equations (where a and b are constants):

−2x1 + 4x2 = 1

4x1 ax2 = b

  • Choose a and b so that the system has a unique solution.
  • Choose a and b so that the system has no solutions.
  • Choose a and b so that the system has many solutions.

Let’s row reduce: −.

Math midterm 1 solution Image 3
  • For a unique solution, choose any a and b so that both the entries that contain a and b are non-zero (so that our system has 2 pivots); a = b = 0 will work.
  • For no solutions, choose a = 8 and any b such that 2 + b , 0; b = 0 will work (the second equation would then be 0x1+ 0x2 = 2, which has no solution).
  • For many solutions, choose a = 8 and b =−2, making the final row zero, thus giving us one free variable for this system.
Want latest solution of this assignment

Submit Your Assignment Here

AssignmentHippo Features

On Time Delivery

Our motto is deliver assignment on Time. Our Expert writers deliver quality assignments to the students.

Plagiarism Free Work

Get reliable and unique assignments by using our 100% plagiarism-free.

24 X 7 Live Help

Get connected 24*7 with our Live Chat support executives to receive instant solutions for your assignment.

Services For All Subjects

Get Help with all the subjects like: Programming, Accounting, Finance, Engineering, Law and Marketing.

Best Price Guarantee

Get premium service at a pocket-friendly rate at AssignmentHippo


  • Assignment Writing Guide
  • Essay Writing Guide
  • Dissertation Writing Guide
  • Research Paper Writing Guide


  • Accounts
  • Computer Science
  • Economics
  • Engineering

Client Review

I was struggling so hard to complete my marketing assignment on brand development when I decided to finally reach to the experts of this portal. They certainly deliver perfect consistency and the desired format. The content prepared by the experts of this platform was simply amazing. I definitely owe my grades to them.

Tap to Chat
Get instant assignment help