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Math 2568 - Midterm 1
Name:
Spring 2014
Oguz Kurt
Problem 1
(20 pts) For which values of k does the following system have
many solutions? Show your work!
x + y + kz =
x + ky + z =
kx + y + z =
no solution, unique solution or infinitely
1
1
1
Problem 2
(15 pts) Show whether the following vectors are linearly independent or not:
       
1
1
1
1
1 0 2 1
       
1 , 1 , 2 , 2
       
2 1 1 0
0
0
0
0
Problem 3
(20 pts) For the following problems, show whether W is a subspace of the vector space V.
)
*
'(
a 0
(a) V = M2×2 , W =
: a, b ∈ R
0 b
 

 x

(b) V = R3 , W = y  : xyz = 0


z
Problem 4
(15 pts) Find a basis for the subspace W = Span
matrices and calculate the dimension of W.
'(
1
0
) (
1
1
,
0
0
) (
0
0
,
−1
0
) (
1
1
,
1
1
)*
0
of the space of 2 × 2
0
Problem 5
1
2
(15 pts) Describe W = a0 + a1 x + a2 x2 + a3 x3 : a0 + 2a1 = a2 + 3a3 as a span of vectors. You do not
need to show whether the spanning vectors are linearly independent or not.
Problem 6
      
 
1
0 
x
 1
(15 pts) Let B = 1 , 0 , 1 be a basis for R3 . Find the coordinate vector y  of the general


1
1
1
z B
 
x
vector y  with respect to B.
z
NOTES:
1. I will not ask about vector space axioms. If I ask them
whether something is a vector space, they should realize that
it is a subset of a vector space and test subspace axioms.
2. I will ask them one problem where they will not be able to
do anything but use basic definitions of span and linear
independence. EX: HW 4, problem 13.
3. While they are allowed to use a calculator, it will probably
be of no use!
4. They are not allowed to use anything that I have not
covered. Example: They cannot use determinant to decide
some stuﬀ.
5. My oﬃce hours are cancelled due to health problems.
6. Exam is during class on Wednesday.
7. I will not give them any formulas.
8. I will ask some problems of the sort: "TRUE or FALSE!
Explain your reasoning."
9. I will not ask them any Hot Chocolate Method problems.