one? 37. Suppose T and U are linear transformations from R" to R" such that T (UX) = x for all x in R". Is it true that U(Tx) = x for all x in R"? Why or why not?
one? 37. Suppose T and U are linear transformations from R" to R" such that T (UX) = x for all x in R". Is it true that U(Tx) = x for all x in R"? Why or why not?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
37
![Invertible Matrix Theorem, expl
Ax = b must have a solution for each b in R".
In Exercises 33 and 34, T is a linear transformation from R2 into
R2. Show that T is invertible and find a formula for 7-¹.
33. T(x₁, x₂) = (-5x₁ + 9x2,4x1 - 7x2)
34. T(x₁, x₂) = (6x₁8x2, -5x1 + 7x₂)
X2)
35. Let T: "R" be an invertible linear transformation. Ex-
plain why T is both one-to-one and onto R". Use equations
(1) and (2). Then give a second explanation using one or more
theorems. lo enm
36. Let T be a linear transformation that maps R" onto R". Show
that T exists and maps R" onto R". Is T-¹ also one-to-
-1
-1
one?
X
37. Suppose I and U are linear transformations from R" to R"
such that T(Ux) = x for all x in R". Is it true that U(TX) = x
for all x in R"? Why or why not?
T
38. Suppose a linear transformation 7 : R" → R" has the prop-
erty that T(u) = T(v) for some pair of distinct vectors u and
v in R". Can T map R" onto R"? Why or why not?
nottoto
till som BR
Pups parl
n
39. Let T: R" → R" be an invertible linear transformation,
and let S and U be functions from R" into R" such that
S (T(x)) = x and U (T(x)) = x for all x in R". Show that
U(v) = S(v) for all v in R". This will show that I has a
unique inverse, as asserted in Theorem 9. [Hint: Given any
v in R", we can write v = T(x) for some x. Why? Compute
S(v) and U(v).]
On toivre sur
40. Suppose T and S satisfy the invertibility equations (1) and
(2), where T is a linear transformation. Show directly that
S is a linear transformation. [Hint: Given u, v in R", let
x = S(u), y = S(v). Then T(x) = u, T(y) = v. Why? Apply
S to both sides of the equation T(x) + T(y) = T(x + y).
Also, consider T(cx) = cT(x).]
c. Use your m
ber of the c
Exercises 42-44 sh
trix A to estimate th
If the entries of A
and if the conditio
positive integer), t
usually be accurate
42. [M] Find the
Construct ar
Then use you
of Ax= b. T
the number c
and report he
used in place
43. [M] Repeat
44.
[M] Solve a
column of th
A =
1
1/2
1/3
1/4
1/5
How many
correct? Ex
56700,-88
45. [M] Some
mand to cre
use an inve
order or la
what you fi
Master
10
11
SOLUTIONS TO PRACTICE PROBLEM
1. The columns of A are ohvioual
SG](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4f7aff33-da2d-447a-b7cd-4f086825b0f0%2F048400ed-e679-4429-aa9d-a1db135dfb93%2Fk13sgpi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Invertible Matrix Theorem, expl
Ax = b must have a solution for each b in R".
In Exercises 33 and 34, T is a linear transformation from R2 into
R2. Show that T is invertible and find a formula for 7-¹.
33. T(x₁, x₂) = (-5x₁ + 9x2,4x1 - 7x2)
34. T(x₁, x₂) = (6x₁8x2, -5x1 + 7x₂)
X2)
35. Let T: "R" be an invertible linear transformation. Ex-
plain why T is both one-to-one and onto R". Use equations
(1) and (2). Then give a second explanation using one or more
theorems. lo enm
36. Let T be a linear transformation that maps R" onto R". Show
that T exists and maps R" onto R". Is T-¹ also one-to-
-1
-1
one?
X
37. Suppose I and U are linear transformations from R" to R"
such that T(Ux) = x for all x in R". Is it true that U(TX) = x
for all x in R"? Why or why not?
T
38. Suppose a linear transformation 7 : R" → R" has the prop-
erty that T(u) = T(v) for some pair of distinct vectors u and
v in R". Can T map R" onto R"? Why or why not?
nottoto
till som BR
Pups parl
n
39. Let T: R" → R" be an invertible linear transformation,
and let S and U be functions from R" into R" such that
S (T(x)) = x and U (T(x)) = x for all x in R". Show that
U(v) = S(v) for all v in R". This will show that I has a
unique inverse, as asserted in Theorem 9. [Hint: Given any
v in R", we can write v = T(x) for some x. Why? Compute
S(v) and U(v).]
On toivre sur
40. Suppose T and S satisfy the invertibility equations (1) and
(2), where T is a linear transformation. Show directly that
S is a linear transformation. [Hint: Given u, v in R", let
x = S(u), y = S(v). Then T(x) = u, T(y) = v. Why? Apply
S to both sides of the equation T(x) + T(y) = T(x + y).
Also, consider T(cx) = cT(x).]
c. Use your m
ber of the c
Exercises 42-44 sh
trix A to estimate th
If the entries of A
and if the conditio
positive integer), t
usually be accurate
42. [M] Find the
Construct ar
Then use you
of Ax= b. T
the number c
and report he
used in place
43. [M] Repeat
44.
[M] Solve a
column of th
A =
1
1/2
1/3
1/4
1/5
How many
correct? Ex
56700,-88
45. [M] Some
mand to cre
use an inve
order or la
what you fi
Master
10
11
SOLUTIONS TO PRACTICE PROBLEM
1. The columns of A are ohvioual
SG
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