Problem Set 1 (due Friday September 3)
MATH 110: Linear Algebra
PART 1
The following problems are each worth 5 points..
1. Curtis p. 14 1.
2. Curtis p. 15 4.
3. Curtis p. 25 3.
4. Curtis p. 33 3.
5. Curtis p. 37 4.
PART 2
Remember that the starred problem is non collaborative.
Problem 1 (5)
Let V=R+, the set of positive real numbers. Define the “sum” of two
elements x, y in Vto be their product xy (in the usual sense), and define
“multiplication of an element xin Vby a scalar cto be xc. Prove that Vis
a real linear space with 1 as the zero element.
Problem 2 (10)
Let Vbe the vector space of all real valued functions defined on the real
line (F(R)). Consider the nexponential functions:
u1(x) = ea1x, . . . , un(x) = eanx
where a1, a2, . . . , anare distinct real numbers. Show that these nfunctions
are independent.
Problem 3∗(5)
Curtis p. 26 10.
In order to do this problem you will have to read some definitions and theo-
rems in Curtis (3.7, 3.9, 3.10 and 3.11).
Problem 4 (15)
Let Vbe a finite dimensional vector space and let Sbe a subspace of V.
Prove each of the following statements:
a) Sis finite dimensional and dim S ≤dim V .
b) dim S =dim V if and only if S=V.
c) Every basis for Sis part of a basis for V.
d) A basis for Vneed not contain a basis for S.
e) Is the union of two subspaces always a subspace? Explain.
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