Scribd
Upload
15 views2 pages

E1 Practice Problems

The document is a review guide for Exam 1 of MA 200 Spring 2024, containing practice problems on linear algebra topics including finding general solutions, determining linear independence, matrix forms, and linear transformations. It includes specific problems such as finding parametric vector forms, examples of echelon forms, and evaluating the consistency of systems of equations. Additionally, it covers concepts like spanning sets and the properties of linear transformations.

Uploaded by

mory yi
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
15 views2 pages

E1 Practice Problems

The document is a review guide for Exam 1 of MA 200 Spring 2024, containing practice problems on linear algebra topics including finding general solutions, determining linear independence, matrix forms, and linear transformations. It includes specific problems such as finding parametric vector forms, examples of echelon forms, and evaluating the consistency of systems of equations. Additionally, it covers concepts like spanning sets and the properties of linear transformations.

Uploaded by

mory yi
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
You are on page 1/ 2

MA 200 Spring 2024 Exam 1 Review - Practice Problems

1. Find the general solution to

x1 − 3x2 − 9x3 + 5x4 = 0


x2 − 2x3 − x4 = 0

and write it in parametric vector form. What is a geometric description of the solution set?

2. Which of the following sets are linearly independent? Justify/explain your answers.
   
 10 5 
(a) −6 , −3
 
2 1
 

     
 1 0 3 
(b) −2 , 0 , −5
   
−1 0 4
 

       

 1 −2 1 0 
       
1 1 −1
, , , 9 

(c)  0   3   2  −4

 
−1 5 3 4
 

3. Give an example of a matrix which is...

(a) ...in echelon form.


(b) ...in reduced echelon form.
(c) ...is not in echelon form.

 
1 1
4. Let A = −2 −1 and T : R2 → R3 by T (⃗x) = A · ⃗x.
−1 −3

(a) What is the domain of T , and what is the codomain of T ?


 
2
(b) For ⃗v = −7, is ⃗v in the range of T ? If so, how many vectors in R2 are there with ⃗v
4
as their image?
5. Provide an argument to explain the following:

“If a set of vectors {v⃗1 , v⃗2 , . . . , v⃗k } in Rn contains more vectors than there are entries in
the vector (i.e., k > n), then the set is linearly dependent.”

 
3 −5 0
6. Let A = −6 2 4 .
9 −7 −4

(a) Do the columns of A span R3 ? If not, find a vector in R3 which is not in the span of
these vectors.

(b) Do the columns of A form a linearly independent set?


(c) Put A into reduced row echelon form.

 
2
7. Let v⃗1 = .
−1

(a) Define a vector v⃗2 such that span{v⃗1 , v⃗2 } = R2 . Explain your reasoning/why your
answer works.
(b) Is your set {v⃗1 , v⃗2 } linearly independent, or linearly dependent? Justify your answer.
(c) Define a vector v⃗3 such that span{v⃗1 , v⃗3 } ̸= R2 . Explain your reasoning/why your
answer works.
(d) Is your set {v⃗1 , v⃗3 } linearly independent, or linearly dependent? Justify your answer.

8. Determine whether each statement is True or False. Justify your response.


(a) The echelon form of a matrix A is uniquely determined by A.
(b) The transformation T : R2 → R2 defined by
   
x1 2x1 + 3x2 − 1
T =
x2 x1 − x2
is a linear transformation.

9. Is the following system consistent or inconsistent? Find a solution if one exists.


2x − 2y = −6
x − y + z=1
3y − 2z = −5

10. Let T : R2 → R2 be the linear transformation given by reflection across the line y = x
followed by projection onto the x-axis. Find the standard matrix of T .

You might also like