Page 130 - 35Linear Algebra

130 Matrices

(b) Choose an ordering for the 3 kinds of supplies and use this to
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rewrite f and g as elements of R .

(c) Let L be a manufacturing process that takes as inputs supply
packages and outputs two products (doors, and door frames). Ex-
plain how it can be viewed as a function mapping one vector space
into another.
(d) Assuming that L is linear and Lf is 1 door and 2 frames, and Lg
is 3 doors and 1 frame, find a matrix for L. Be sure to specify
the basis vectors you used, both for the input and output vector
space.

2. You are designing a simple keyboard synthesizer with two keys. If you
push the first key with intensity a then the speaker moves in time as
a sin(t). If you push the second key with intensity b then the speaker
moves in time as b sin(2t). If the keys are pressed simultaneously,

(a) describe the set of all sounds that come out of your synthesizer.
(Hint: Sounds can be “added”.)

3
(b) Graph the function ∈ R {1,2} .
5

3
(c) Let B = (sin(t), sin(2t)). Explain why is not in R {1,2} but
5
B
is still a function.

3
(d) Graph the function .
5
B
3. (a) Find the matrix for d acting on the vector space V of polynomi-
dx
2
als of degree 2 or less in the ordered basis B = (x , x, 1)

(b) Use the matrix from part (a) to rewrite the differential equation
d p(x) = x as a matrix equation. Find all solutions of the matrix
dx
equation. Translate them into elements of V .

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