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? X
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? X
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
![33. T(x₁, x₂) = (-5x₁ + 9x2, 4x1 - 7.x2)
34. T(x₁, x₂) = (6x₁ - 8x2, -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 T-¹ also one-to-
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?
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?
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).]
40. Suppose T and S satisfy the invertibility equations (1) and
(2), where I 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).]
SOLUTIONS TO
Ex
trix
If th
and
pos
usu
42.
43.
44.
45.
SG](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8eddd3e4-97b2-4f20-b642-1ed0827b897f%2F3cf66b57-fc9a-407c-b54a-dbc094bff1a8%2F8lvbqxf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:33. T(x₁, x₂) = (-5x₁ + 9x2, 4x1 - 7.x2)
34. T(x₁, x₂) = (6x₁ - 8x2, -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 T-¹ also one-to-
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?
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?
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).]
40. Suppose T and S satisfy the invertibility equations (1) and
(2), where I 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).]
SOLUTIONS TO
Ex
trix
If th
and
pos
usu
42.
43.
44.
45.
SG
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