Math 3130
Abrams
Spring 2016
PRACTICE EXAM for Exam 1
Give complete answers. Show all your work!! Write legibly. Write out your explanations in full detail.
Circle answers when appropriate. You may use the back of the sheet if necessary. Good luck.
1. SHORT ANSWER
(a) (2 pt) If V is a vector space and S = {v1 , v2 , ..., vn } is a finite set of vectors in V , then S is called a
basis for V in case
(i)
and
(ii)
(b) (1 pt) Referring to question 1a, the number n is called the
of V .
(c) (1 pt) Give an example of a vector space V which has dim(V ) = 4, but where V is NOT R4 .
(d) (3 pt) Do the vectors (1, 2, 3), (1, 0, 0), and (4, 4, 6) form a basis of R3 ? Justify your answer appropriately.
2. (3 pt) Is the vector u = (3, −1, 3, 2) in the span of the vectors v1 = (1, 0, 0, 0), v2 = (1, −1, 1, 0), and
v3 = (0, 0, 1, 1)? (Rephrased: Is u in span{v1 , v2 , v3 }?) If so, write u as a linear combination of v1 , v2 , and v3 .
If not, explain why not by an appropriate computation.
1
3. (3 pt) Consider the set S of vectors in P2 of the form a + bx + cx2 where a ≥ 0. S is NOT a subspace of
P2 . Explain why not.
4. (7 pt) Let W denote the subset of P3 consisting of all polynomials of the form a0 + a1 x + a2 x2 + a3 x3 for
which 2a0 − a1 + a3 = 0.
(a) Give three specific examples of polynomials in P3 which are in the subset W .
(b) Give three specific examples of polynomials in P3 which are NOT in the subset W .
(c) Prove that W is a subspace of P3 . (As in the homework, use The Subspace Theorem.) Make sure to
show all your work, and that your notation is clear and precise.
5. (4 pt) It can be easily shown that the set of vectors v1 = (1, 0, 0), v2 = (−1, 1, 0), v3 = (0, 1, 2) forms a
basis for R3 . Find the coordinate vector of u = (1, 1, 3) relative to this basis.
2


2 0 −1
6. (11 pt total) Consider the matrix A =  1 1 0  .
4 0 −2
(a) Find a basis for the null space of A.
(b) Find a basis for the column space of A.
(c) Use your answer to part (b) to find the dimension of the row space of A. (Do NOT find a basis of
the row space of A.)
(d) For any m × n matrix M , what is the equation which arises in the Dimension Theorem for Matrices?
(e) For the specific matrix A of this question, give the specific values which arise in the Dimension
Theorem for Matrices.
3
7. (9 pt total) Let V be the vector space F(R),Ri.e., the set of functions from R to R. Let W denote the subset
1
of V consisting of all those functions f for which −1 f (x)dx = 0.
(a) Is the function f (x) = x5 in W ? Explain why or why not.
(b) Prove that W is a subspace of V . (Use the Subspace Theorem.)
(c) Verify that the set consisting of the two functions {x, x3 } is a linearly independent subset of W .
(d) What is the dimension of W ?
8.
(1 pt each) TRUE / FALSE.
(a)
T
F
If S is a set of 3 vectors in R3 , then S must be a basis of R3 .
(b)
T
F
If S is a set of 3 vectors in R4 , then S cannot be a basis of R4
(c) T
F If you and your friend each find a basis for a vector space V , then you will each necessarily
have found the same number of vectors.
(d) T
F
The function sin x is a vector.
(e) T
F
Any set of 2 vectors in P4 is a linearly independent set.
(f)
T
F
Let S be a linearly independent set of 3 vectors in R3 . Then S is necessarily a basis for
(g) T
F
A basis for a vector space is a subspace of the vector space.
(h) T
F
For any matrix A, the row space of A and the null space of A have the same dimension.
3
R.
(i) T
F
If a set of vectors S in a vector space V has the property that no vector in S is a scalar
multiple of any other vector in S, then S is a linearly independent subset of V.
4