Given a 2 x 2 matrix A, let u, v be the solutions of ú= Au and u(0) = -H Define the (time-dependent) matrix Þ(t) = [u(t)_v(t)]. (a) Show that the solution of x = Ax x(0) = X0 is given by x(t) = $(t)xo. = v = Av (0) - = v(
Given a 2 x 2 matrix A, let u, v be the solutions of ú= Au and u(0) = -H Define the (time-dependent) matrix Þ(t) = [u(t)_v(t)]. (a) Show that the solution of x = Ax x(0) = X0 is given by x(t) = $(t)xo. = v = Av (0) - = v(
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.2: Linear Independence, Basis, And Dimension
Problem 3AEXP
Related questions
Question
Need b and c
![Given a 2 x 2 matrix A, let u, v be the solutions of
ú= Au
and
u(0)
H
8
Define the (time-dependent) matrix Þ(t) = [u(t) _v(t)].
(a) Show that the solution of
x = Ax
x(0) =
= XO
is given by
x(t) = $(t)xo.
(b) Find Þ(t) for each of the three real canonical forms.
(c) Suppose that B = PMP-1 for an invertible 2 × 2 matrix P. Show that the solution of
x = Bx
x(0) = Xo
is given by
x(t) = PÞ(t)P−¹xo-
(d) For each of the following matrices A, find a matrix M so that A
PMP-1, where M is one of
=
the real canonical forms above, and P
=
11.
Then apply your answers to parts (b) and (c) to
1 2
find the corresponding Þ(t).
5 4
(i) A=
-2
1
5
=
-2 -5
-2 4
(ii) A
(iii) A =
- 1
-6
=
v = Av
v(0)
=](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffd81632b-b840-4145-b4b6-92f20885be52%2Fdc1c7367-a7bc-4477-8c83-a6b4522ce031%2Ff2hj64e_processed.png&w=3840&q=75)
Transcribed Image Text:Given a 2 x 2 matrix A, let u, v be the solutions of
ú= Au
and
u(0)
H
8
Define the (time-dependent) matrix Þ(t) = [u(t) _v(t)].
(a) Show that the solution of
x = Ax
x(0) =
= XO
is given by
x(t) = $(t)xo.
(b) Find Þ(t) for each of the three real canonical forms.
(c) Suppose that B = PMP-1 for an invertible 2 × 2 matrix P. Show that the solution of
x = Bx
x(0) = Xo
is given by
x(t) = PÞ(t)P−¹xo-
(d) For each of the following matrices A, find a matrix M so that A
PMP-1, where M is one of
=
the real canonical forms above, and P
=
11.
Then apply your answers to parts (b) and (c) to
1 2
find the corresponding Þ(t).
5 4
(i) A=
-2
1
5
=
-2 -5
-2 4
(ii) A
(iii) A =
- 1
-6
=
v = Av
v(0)
=
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