Math 441/541 Assignment 1: Solutions
Chapter 1 3. Here we use the associativity of scalar multiplication and the fact that 1 v = v . (v ) = 1 (v ) = 1 (1 v ) = (1 1)v = 1 v = v.
4. Let a F and v V and suppose av = 0. If a = 0 then were done. So assume a =
Group
Names
Group Quiz for Section 1.3
1
1
3 and 0 in R3 ? Justify your answer.
What is the span of
0
0
Note: to prove two sets are equal, you have to prove they are exactly the same; that
is, that each is a subset of the other.
Group
Names
Group Quiz for Section 1.5
Let A Mn (F) (that is, A is an nn matrix over F). The trace of A = [aij ]1i,jn
is dened to be
n
tr(A) :=
aii .
i=1
Show that the set
cfw_A Mn (F) : tr(A) = 0
is a subspace of Mn (F).
Group
Names
Group Quiz for Section 1.2
Determine whether each of the matrices below is in row-echelon form, reduced
row-echelon form, or neither. For any matrices in RREF, solve the corresponding
system or explain why the system is inconsistent.
1 0 0
1 3
Math 307 Homework
August 26, 2015
1. Suppose that f (x) = ax2 + bx + c is a quadratic polynomial whose graph passes
through the points (1, 1), (0, 0), and (1, 2). Use this information to write down
a linear system satised by (a, b, c), and nd a solution f
Group
Names
Group Quiz for Section 1.6
Let C(R) be the vector space of all continuous functions f : R R. For f C(R),
dene T f : R R by (T f )(x) = f (x) cos x.
Show that T is a linear map from C(R) to C(R).
Math 307 Homework
August 24, 2015
1. Suppose that x1 = c1 , . . . , xn = cn is a solution of the linear system
a11 x1 + + a1n xn = b1 ,
.
.
.
am1 x1 + + amn xn = bm .
Under what circumstances is x1 = 2c1 , . . . , xn = 2cn also a solution?
2. Give a geome
Names
Group Quiz for Section 1.4
We use F3 to stand for the set cfw_0, 1, 2 with the following addition and multiplication tables:
+
0
1
2
0
0
1
2
1
1
2
0
2
2
0
1
0
1
2
0
0
0
0
1
0
1
2
2
0
2
1
Which element (0, 1, or 2) is each of the following equal to (
Group
Names
Group Quiz for Section 1.4
We use F3 to stand for the set cfw_0, 1, 2 with the following addition and multiplication tables:
+
0
1
2
0
0
1
2
1
1
2
0
2
2
0
1
0
1
2
0
0
0
0
1
0
1
2
2
0
2
1
Which element (0, 1, or 2) is each of the following equa