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? RAUD ale
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? RAUD ale
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
![ing
tion Ax = 0 has only the trivial solution. Without
Invertible Matrix Theorem, explain directly why the equation
= b must have a solution for each b in R".
Ax:
In Exercises 33 and 34, 7 is a linear transformation from R2 into
R2. Show that T is invertible and find a formula for T-¹.
33. T(x₁, x₂) = (-5x₁ + 9x2,4x₁ - 7x₂)
34.
T(x₁, x₂) = (6x₁8x2, -5x₁ + 7x2)
35.
Let T: R"→ 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.
-
36. Let T be a linear transformation that maps R" onto R". Show
that T exists and maps R" onto R". Is T¹ also one-to-
one?
n
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?
38. Suppose a linear transformation T: 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?
n
39. Let T: R" → R" be an invertible linear transformation,
and let S and U be functions from R" into Rn 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).]
Compro
b. The
.05%
of (4
c. Use:
ber o
Exercises 42-
trix A to estim
If the entries
and if the con
positive integ
usually be acc
42. [M] Find
Construc
Then use
of Ax =
the numb
and repo
used in P
43. [M] Rep
44.
[M] Solv
column c
1,
A = 1,
1,
How man
correct? E
56700,-:
40. Suppose T and S satisfy the invertibility equations (1) and
S is a linear transformation. [Hint: Given u, v in R", let
(2), where T is a linear transformation. Show directly that 45. [M] Som
mand to c
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).]
use an inv
order or
what you](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F046862d2-b2b1-4a60-bae0-248c79b95a0e%2F868e2fb8-ca97-4264-a1aa-f451667c2157%2Fcj8goq_processed.jpeg&w=3840&q=75)
Transcribed Image Text:ing
tion Ax = 0 has only the trivial solution. Without
Invertible Matrix Theorem, explain directly why the equation
= b must have a solution for each b in R".
Ax:
In Exercises 33 and 34, 7 is a linear transformation from R2 into
R2. Show that T is invertible and find a formula for T-¹.
33. T(x₁, x₂) = (-5x₁ + 9x2,4x₁ - 7x₂)
34.
T(x₁, x₂) = (6x₁8x2, -5x₁ + 7x2)
35.
Let T: R"→ 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.
-
36. Let T be a linear transformation that maps R" onto R". Show
that T exists and maps R" onto R". Is T¹ also one-to-
one?
n
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?
38. Suppose a linear transformation T: 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?
n
39. Let T: R" → R" be an invertible linear transformation,
and let S and U be functions from R" into Rn 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).]
Compro
b. The
.05%
of (4
c. Use:
ber o
Exercises 42-
trix A to estim
If the entries
and if the con
positive integ
usually be acc
42. [M] Find
Construc
Then use
of Ax =
the numb
and repo
used in P
43. [M] Rep
44.
[M] Solv
column c
1,
A = 1,
1,
How man
correct? E
56700,-:
40. Suppose T and S satisfy the invertibility equations (1) and
S is a linear transformation. [Hint: Given u, v in R", let
(2), where T is a linear transformation. Show directly that 45. [M] Som
mand to c
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).]
use an inv
order or
what you
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