f(t) = 1-8 cos? t + 8 cos 1 g(1) = --1+ 18 cos? t- 48 cos't + 32 cos ! %3D Study the graph of f for 0
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
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CHAPTER 4
Vector Spaces
35. [M) Show that w is in the subspace of R spanned by
V1, V2, V3, where
in a later exercise:
f(t) = 1-8 cos? t+8 cos' t
g(t) = --1 + 18 cos?t-48 cos“ t + 32 cos 1
8
-4
-7
W =
3
, V3 =
V =
V2 =
Study the graph of f for 0 <t < 27, and guess a simple for-
mula for f(t). Verify your conjecture by graphing the differ-
ence between 1 + f(1) and your formula for f(t). (Hopefully,
you will see the constant function 1.) Repeat for g.
-2
-5
9
36. [M] Determine if y is in the subspace of R“ spanned by the
columns of A, where
-4
38. [M] Repeat Exercise 37 for the functions
3 -5 -9
7 -6
-8
8
y =
A =
f(1) = 3 sint - 4 sin³ t
-5 -8
3
g(t) = 1- 8 sin? t + 8 sin" t
h(t) = 5 sin t – 20 sin' t + 16 sin³ t
-5
2 -2 -9
37. [M] The vector space H = Span {1, cos? t, cos“ t, cos 1}
contains at least two interesting functions that will be used
in the vector space Span {1, sint, sin? 1,..., sin t}.
SOLUTIONS TO PRACTICE PROBLEMS
1. Take any u in H-say,u =
-and take any c+1-say, c= 2.
Then
cu =
If this is in H , then there is some s such that
3s
2+ 5s
That is, s = 2 and s = 12/5, which is impossible. So 2u is not in H and H is not a
vector space.
BURE
2. V1 = 1v1 + 0v2 + + 0vp.
V1,...,Vp, so Vị is in W. In general, v is in W because
This expresses vi as a linear combination of
EXAMPLE 1 V = 0v1 + + Ovk-1 +1v + Ovk+1 + · … + 0v,
belos to the mll spec
4.2 NULL SPACES, COLUMN SPACES, AND LINEAR TRANSFORMATIONS
In applications of linear algebra, subspaces of R" usually arise in one of two ways: (1) as
the set of all solutions to a
of all linear combinations of certain specified vectors. In this section, we compare and
contrast these two descriptions of subspaces, allowing us to practice using the concept of
a subspace. Actually, as you will soon discover, we have been working with subspaces
ever since Section 1.3. The main new feature here is the terminology. The section
concludes with a discussion of the kernel and range of a linear transformation.
tem of homogeneous linear equations or (2) as the
The Null Space of a Matrix
Consider the following system of homogeneous equations:
X1 - 3x2 - 2x3 = 0
-5x1 + 9x2 + X3 = 0
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