linear-guest

130 Matrices
(b) Choose an ordering for the 3 kinds of supplies and use this to
rewrite f and g as elements of R 3 .
(c) Let L be a manufacturing process that takes as inputs supply
packages and outputs two products (doors, and door frames). Explain
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
5)
{1,2} .
(c) Let B = (sin(t), sin(2t)). Explain why
is still a function.
( 3
(d) Graph the function
5)
.
B
( 3
5)
B
is not in R {1,2} but
3. (a) Find the matrix for d acting on the vector space V of polynomials
of degree 2 or less in the ordered basis B = (x 2 , x,
dx
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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