Download 8 – 10 Let CR denote the set of continuous functions on R (this is a

HW 11
8 – 10
Let CR denote the set of continuous functions on R (this is a vector space using function
addition and scalar multiplication). Define L : CR → FR by
Z x
L(f )(x) =
f (t) dt.
0
Show that L is linear.
8 – 11
Let S be the subset of FR consisting of all functions f for which f (−x) = f (x) for all
x ∈ R. Show that S is a subspace of FR .
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8 – 12
Let L : M2×2 → P2 be the linear function given by
a b
L
= (a − d)x + (b + c).
c d
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(a) Find ker L.
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(b) Find im L.
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8 – 13
Let L : V → V ′ be a linear function, let a ∈ V ′ and let vp be a particular vector in V such
that L(vp ) = a.
(a) Show that if v0 is in ker L, then L(vp + v0 ) = a.
(b) Show that if v is a vector in V such that L(v) = a, then v = vp + v0 for some
v0 ∈ ker L. (Hint: If you show that v − vp ∈ ker L, then you can conclude that
v − vp = v0 for some v0 ∈ ker L.)
(c) Recall that L−1 (a) is the set of all v in V such that L(v) = a. Show that L−1 (a) =
{vp + v0 | v0 ∈ ker L}. (Hint: Use parts (a) and (b) to show that each set is contained
in the other.)
8 – 14
Let L : DR → FR be the function given by
L(y) = y ′ + 3y
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(DR is the vector space of differentiable functions).
(a) Show that L is linear.
(b) Let yp be a particular solution to the linear differential equation y ′ + 3y = 1. Show
that every solution of y ′ + 3y = 1 is of the form y = yp + y0 where y0 is a solution
to the corresponding homogeneous equation y ′ + 3y = 0. (Hint: Use part (a) and
Exercise 8 – 13(b).)
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8 – 15
8 – 16
5 6
In the vector space M2×2 , determine whether
is in
1 −5
1 −3
−2 6
2
1
Span{
,
,
}.
−4 −1
8 2
−1 −2
Let
4
8
3
, g(x) = 2 − , h(x) = 1 + .
x
x
x
Determine whether the functions f, g, and h are linearly independent in the vector space
FR+ , where R+ is the set of positive real numbers.
f (x) = 1 +
Hint: At some point, make three choices for x to get a system of three equations.
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