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May 23rd

May 23rd

# VECTOR SPACES

MATH 294 FALL 1982 PRELIM 1 # 3a
2.3.1
Let C[0, 1] denote the space of continuous functions defined on the interval [0,1]
(i.e. f(x) is a member of C[0, 1] if f(x) is continuous for 0 ≤ x ≤1). Which one of
the fol lowing subsets of C[0, 1] does not form a vector space? Find it and explain
why it does not.

MATH 294 SPRING 1982 PRELIM 1 # 3
2.3.2
a)
i) The subset of functions f which belongs to C[0, 1] for which .
ii) The set of functions f in C[0, 1] which vanish at exactly one point (i.e. f(x) = 0
for only one x with 0 ≤ x ≤1). Note different functions may vanish at different
points within the interval.
iii) The subset of functions f in C[0, 1] for which f(0) = f(1).
b) Let f(x) = x3 + 2x + 5. Consider the four vectors
,, f' means .
i) What is the dimension of the space spanned by the vectors? Justify your answer.
ii) Ex press x 2 + 1 as a linear combination of the .

MATH 294 FALL 1984 FINAL # 1
2.3.3
a) De termine which of the following subsets are subspaces of the indicated vector
spaces, and for each subspace determine the dimension of th- space. Explain your
i) The set of all vectors in R2 with first component equal to 2.
ii) The set of all vectors in R3 for which .
iii) The set of all vectors in R3 satisfying .
iv) The set of all functions f(x) in C[0, 1] such that . Recall that
C[0, 1] denotes the space of all real valued continuous functions defined on the
closed interval [0,1].
b) Find the equation of the plane passing through the points (0,1,0), (1,1,0) and
(1,0,1), and find a unit vector normal to this plane.

MATH 294 SPRING 1985 FINAL # 5
2.3.4
Vectors and both lie in Rn. The vector
a) Also lies in Rn.
b) May or may not lie in Rn.
c) Lies in
d) Does not lie in Rn.

MATH 294 SPRING 1985 FINAL # 6
2.3.5
The vector space Rn
a) Contains the zero vector.
b) May or may not contain the zero vector.
c) Never contains the zero vector.
d) Is a complex vector space.

MATH 294 SPRING 1985 FINAL # 7
2.3.6
Any set of vectors which span a vector space
a) Always contains a subset of vectors which form a basis for that space.
b) May or may not contain a subset of vectors which form a basis for that space.
c) Is a linearly independent set.
d) Form an orthonormal basis for the space.

MATH 294 SPRING 1987 PRELIM 3 # 7
2.3.7
For what values of the constant a are the functions {sin t and sin(t + a)} in
linearly independent?

MATH 294 SPRING 1987 PRELIM 3 # 9
2.3.8
For problems (a) - (c) use the bases B and B' below:
and
a) Given that what is ?
b) Using the standard relation between R2 and points on the plane make a sketch
with the point clearly marked. Also mark the point , where .
c) Draw the line defined by the points and . Do the points on this line represent
a subspace of R2?

MATH 294 FALL 1987 PRELIM 3 # 2
2.3.9
In parts (a) - (g) answer “true” if V is a vector space and “false” if it is not (no
partial credit):
a) V = set of all x(t) in such that x(0) = 0.
b) V = set of all x(t) in such that x(0) = 1.
c) V = set of all x(t) in such that (D + 1)x(t) = 0.
d) V = set of all x(t) in such that (D + 1)x(t) = et.
e) V = set of all polynomials of degree less than or equal to one with real coefficients .
f ) V = set of all rational numbers (a rational number can be written as the ratio of
two integers , e.g., is a rational number while π = 3.14 . . . is not)
g) V = set of all rational numbers with the added restriction that scalars must also
be rational numbers.

MATH 294 FALL 1987 FINAL # 7
2.3.10
Consider the boundary-value problem
X'' + λX = 0 0 < x < π, X(0) = X(π ) = 0, where λ is a given real number.
a) Is the set of all solutions of this problem a subspace of ? Why?
b) Let W = set of all functions X(x) in such that X(0) = X( π) = 0. Is
T ≡ D2 −λ linear as a transformation of W into ? Why?
c) For what values of is Ker(T) nontrivial?
d) Choose one of those values of λ and determine Ker(T)

MATH 293 SPRING 1990? PRELIM 2 # 3
2.3.11 Is the set of vectors in linearly independent or dependent?

MATH 293 SPRING 1992 PRELIM 2 # 3
2.3.12
W is the subspace of spanned by the vectors ,Find
the dimension of W and give a basis.

MATH 293 SPRING 1992 PRELIM 2 # 4
2.3.13
V is the vector space consisting of vector-valued functions where
and are continuous functions of t in 0 ≤ t ≤1. W is the subset of V
where the functions satisfy the differential equations
and
Is W a subspace of V ?

MATH 293 SPRING 1992 PRELIM 2 # 6
2.3.14
V is the vector space consisting of all 2×2 matrices . Here the
are arbitrary real numbers and the addition and scalar multiplication are defined
by
and
a) Is a subspace? If so give a basis for .
b) Same as part (a) for .
c) Show that , and are linearly independent.
d) What is the largest possible number of linearly independent vectors in V ?

MATH 293 SPRING 1992 FINAL # 7
2.3.15
A “plane” in means, by definition, the set of all points of the form where
is a constant (fixed) vector and varies over a fixed two-dimensional subspace of
. Two planes are “parallel” if their subspaces are the same. It is claimed that
the two planes:

1st plane:

2nd plane:

(where and can as sume any scalar values) do not intersect and are not
parallel. Do you agree or disagree with this claim? You have to give very clear
reasons for your answer in order to get credit for this problem.

MATH 293 SUMMER 1992 FINAL # 3
2.3.16
a) Let V be the vector space of all 2 matrices of the form

where , i, j = 1, 2, are real scalars.
Consider the set S of all 2times2 matrices of the form

where a and b are real scalars.
i) Show that S is a subspace. Call it W.
ii) Find a basis for W and the dimension of W.
b) Consider the vector space V {f(t) = a + b sin t + c cos t}, for all real scalars a, b
and c and 0 ≤ t ≤1
Now consider a subspace W of V in which at t = 0
Find a basis for the subspace W.

MATH 293 FALL 1992 PRELIM 3 # 3
2.3.17
Let C(−π ,π ) be the vector space of continuous functions on the interval −π ≤
x ≤ π. Which of the following subsets S of C(−π , π) are subspaces? If it is not a
subspace say why. If it is, then say why and find a basis.
Note: You must show that the basis you choose consists of linearly independent
vectors. In what follows and are arbitrary scalars unless otherwise stated.
a) S is the set of functions of the form
b) S is the set of functions of the form , subject to the
condition
c) S is the set of functions of the form , subject to the
condition

MATH 293 FALL 1992 PRELIM 2 # 5
2.3.18
Consider all polynomials of degree ≤ 3

They Form a vector space. Now consider the subset S of consisting of polynomials
of degree ≤ 3 with the conditions

MATH 293 FALL 1992 PRELIM 2 # 6
2.3.19
Given a vector space which is the space of all vectors of the form for
all real
consider the set S of vectors in of the form

for all values of scalars a, b and c.
Is the set S a subspace W of ? Explain your answer carefully.

MATH 293 FALL 1992 FINAL # 3d
2.3.20 Let S be the set of all vectors of the form where , and are the
usual mutually perpendicular unit vectors. Let W be the set of all vectors that are
perpendicular to the vector . Is W a vector subspace of ? Explain

MATH 293 FALL 1994 PRELIM 2 # 5
2.3.21
In each of the following, you are given a vector space V and a subset W. Decide
whether W is a subspace of V , and prove that your answer is correct.
a) V is the space of all 2 × 2 matrices, and W is the set of 2 × 2 matrices A
such that A2 = A
b) V is the space of differentiable functions, and W is the set of those differentiable
functions that satisfy f'(3) = 0.

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