Math 130
Homework #1
Solutions
Problem 1. Solve
x + 2y = 8
3x 4y = 4
Multiply the rst equation by 3 and add it to the second equation.
Replace the second equation with the new equation, leaving the rst
equation unchanged.
x + 2y = 8
10y = 20
Solve the sec
A basis for a vector space. You know some
bases for vector spaces already even if you havent
know them by that name.
For instance, in R3 the three vectors i = (1, 0, 0)
which points along the x-axis, j = (0, 1, 0) which
points along the y-axis, and k = (0
3. [20] Determine if the set
S = cfw_(1, 1, 2), (1, 2, 1), (1, 1, 4)
First Test Answers
Math 130 Linear Algebra
of three vectors in R3 is independent or dependent.
Show your work for credit.
D Joyce, October 2013
Is there a nontrivial solution to the vect
Math 130
Homework #9
Solutions
Section 6.2.
Problem 5.
(a) Given the set V = cfw_(a, b, c) R3 | c = 2. Since
(a, b, 2) + (a , b , 2) = (a + a , b + b , 4) V
it follows that V is not subspace of R3 .
(b) Given the set V = cfw_(a, b, c) R3 | c = a + b. Then
Math 130
Homework #8
Solutions
Section 4.3
Problem 1.
(a) Given L : R2 R3 dened by L(x, y) = (x + 1, y, x + y). For
all k R,
L(k(x, y) = L(kx, ky) = (kx + 1, ky, kx + ky)
kL(x, y) = k(x + 1, y, x + y) = (kx + k, ky, kx + ky)
L is not linear, since L(k(x,
Chapter 1 Review Questions: Solutions
1. What do we mean when we say a system of equations is linear? What do we mean
when we say a function f is linear, i.e., satises the linearity property? For example, matrix multiplication satises the linearity proper
Chapter 1 Review Questions
1. What do we mean when we say a system of equations is linear? What do we mean
when we say a function f is linear, i.e., satises the linearity property? For example, matrix multiplication satises the linearity property.
2. True
the identity function on B. The usual notation for
the function inverse to f is f 1 .
If f and g are inverse to each other, that is, if g
is the inverse of f , g = f 1 , then f is the inverse of
g, f = g 1 . Thus, (f 1 )1 = f .
An important property of bi