MATH2270: Midterm 1 Practice Problems
The following are practice problems for the first exam.
1. For what values of h and k is the following system consistent?
2x1 − x2 = h
−6x1 + 3x2 = k
Answer: k = −3h, h ∈ R
2. Give a parametric description of the solutions
shown below:

1 −2 3
0 0 0

0 0 0
0 0 0
 −29 
2
" −3 #
1
0
1
0
0
Answer:{x2  00  + x3
0
0
3. Determine if the vector ~b =
+ x5 
h2i
3
1
0
0
−4
1
0
to the equation A~x = ~0 where A is the matrix

−6 5 0
1 4 −6

0 0 1
0 0 0
 | x2 , x3 , x5 ∈ R}
is in Span{v1 , v2 } where
 
1
v~1 = 2 ,
2
 
5
v~2 = 9
7
If the answer is yes, then write ~b as a linear combination of v~1 and v~2 . Answer: Yes. ~b =
−3v~1 + v~2
4. If ~b is in the span of the vectors v~1 , . . . , v~k , what can you say about solutions to the matrix
equation A~x = ~b where A is the matrix whose columns are v~1 , . . . , v~k (i.e., A = [v~1 v~2 . . . v~k ])?
Answer: The equation A~x = ~b is consistent.
h1i h1i h0i
5. Is the set of vectors { 2 , 1 , 0 } linearly independent? Why or why not? Answer: No
3
1
0
1
6. Is the set of vectors {Span{
}} linearly independent? Why or why not? Answer: No
0
7. Determine if the linear transformation T : R3 → R2 , whose standard matrix is A, is 1-1. Is it
onto?
2 1 0
A=
1 1 1
Answer: Yes
8. Suppose S : R2 → R3 is a linear transformation such that S ([ 10 ]) =
h −12 i
−3
(a) Find S
. Answer: 12
3
9
1
h
3
−2
−1
i
and S ([ 01 ]) =
h −1 i
2
2
.
(b) Find the standard matrix for S. Answer:
h
3 −1
−2 2
−1 2
i
9. Be able to multiply matrices...
10. Write down the inverse of the following matrix:
3 2
8 5
Answer:
−5
2
8 −3
11. Compute the inverse of the following matrix:

1
2
−4 −7
−2 −6

−1
3
3
h 3 0 1 i
Answer: −6 −1 −1 . This question is probably more computationally intensive than any−10 −2 −1
thing you will see on the exam.
12. Suppose a linear transformation T : Rn → Rn has the property that T (~u) = T (~v ) for some
pair of distinct vectors ~u and ~v . Can T be onto? Why or why not? No. Something about
invertible matrices.
2