Page 131 - 35Linear Algebra
P. 131
7.2 Review Problems 131
(c) Find the matrix for d acting on the vector space V in the ordered
dx
0
2
2
basis B = (x + x, x − x, 1).
(d) Use the matrix from part (c) 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 .
(e) Compare and contrast your results from parts (b) and (d).
4. Find the “matrix” for d acting on the vector space of all power series
dx
3
2
in the ordered basis (1, x, x , x , ...). Use this matrix to find all power
series solutions to the differential equation d f(x) = x. Hint: your
dx
“matrix” may not have finite size.
5. Find the matrix for d 2 2 acting on {c 1 cos(x) + c 2 sin(x) | c 1 , c 2 ∈ R} in
dx
the ordered basis (cos(x), sin(x)).
6. Find the matrix for d acting on {c 1 cosh(x) + c 2 sinh(x)|c 1 , c 2 ∈ R} in
dx
the ordered basis
(cosh(x), sinh(x))
and in the ordered basis
(cosh(x) + sinh(x), cosh(x) − sinh(x)).
2
7. Let B = (1, x, x ) be an ordered basis for
2
V = {a 0 + a 1 x + a 2 x | a 0 , a 1 , a 2 ∈ R} ,
0
3
2
and let B = (x , x , x, 1) be an ordered basis for
3
2
W = {a 0 + a 1 x + a 2 x + a 3 x | a 0 , a 1 , a 2 , a 3 ∈ R} ,
Find the matrix for the operator I : V → W defined by
Z x
Ip(x) = p(t)dt
1
relative to these bases.
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