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
2. Give a parametric description of the solutions to the equation A~x = ~0 where A is the matrix
shown below:


1 −2 3 −6 5 0
0 0 0 1 4 −6


0 0 0 0 0 1 
0 0 0 0 0 0
h2i
3. Determine if the vector ~b = 3 is in Span{v1 , v2 } where
1
 
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 .
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 ])?
h1i h1i h0i
5. Is the set of vectors { 2 , 1 , 0 } linearly independent? Why or why not?
3
1
0
1
6. Is the set of vectors {Span{
}} linearly independent? Why or why not?
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
h 3 i
h −1 i
8. Suppose S : R2 → R3 is a linear transformation such that S ([ 10 ]) = −2 and S ([ 01 ]) = 2 .
−1
−3
(a) Find S
.
3
(b) Find the standard matrix for S.
9. Be able to multiply matrices...
10. Write down the inverse of the following matrix:
3 2
8 5
1
2
11. Compute the inverse of the following matrix:

1
2
−4 −7
−2 −6

−1
3
3
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?
2