In Exercises 21 and 22, mark each statement True or False. Justify each answer. 21/a. 21 a. A linear transformation is a special type of function. b. If A is a 3 x 5 matrix and T is a transformation defined by T(x) = Ax, then the domain of T is R³. c. If A is an m x n matrix, then the range of the transforma- tion x Ax is Rmami ist d. Every linear transformation is a matrix transformation. e. A transformation T is linear if and only if T(C₁V₁+ C₂V2) = C₁T (V₁) + C₂T (v₂) for all v₁ and v₂ in the domain of T and for all scalars c₁ and C₂. 2 2
In Exercises 21 and 22, mark each statement True or False. Justify each answer. 21/a. 21 a. A linear transformation is a special type of function. b. If A is a 3 x 5 matrix and T is a transformation defined by T(x) = Ax, then the domain of T is R³. c. If A is an m x n matrix, then the range of the transforma- tion x Ax is Rmami ist d. Every linear transformation is a matrix transformation. e. A transformation T is linear if and only if T(C₁V₁+ C₂V2) = C₁T (V₁) + C₂T (v₂) for all v₁ and v₂ in the domain of T and for all scalars c₁ and C₂. 2 2
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
21
![-2
and let
- ()--[3]
V₁ =
-[-3].
5
T: R² → R² be a linear transformation that maps x into
X₁V₁ + X2V₂. Find a matrix A such that T(x) is Ax for
each x.
20. Let x =
X2
and V₂ =
In Exercises 21 and 22, mark each statement True or False. Justify
1 ban
each answer.
21/
21 a. A linear transformation is a special type of function.
b. If A is a 3 x 5 matrix and T is a transformation defined
by T(x) = Ax, then the domain of T is R³.
to 4
lemb1009 1singamos
1091
c.
If A is an m x n matrix, then the range of the transforma-
tion x → Ax is Rm.
d. Every linear transformation is a matrix transformation.
OT DAN DISIqp21
e.
A transformation T is linear if and only if T(C₁V₁+
C2V2) = C₁T(V₁) + C₂T (V₂) for all v₁ and v2 in the
domain of T and for all scalars c₁ and ₂.
22. a. Every matrix transformation is a linear transformation.
b.
The codomain of the transformation x → Ax is the set of
all linear combinations of the columns of A.
n
c. If T: R" → Rm is a linear transformation and if c is
in Rm, then a uniqueness question is "Is c in the range
of T?"
d. A linear transformation preserves the operations of vector
addition and scalar multiplication.
DODENNO
U
linear transformation.
e. The superposition principle is a physical description of a
2
23. Let T: R² → R2 be the linear transformation that reflects
each point through the x₁-axis. (See Practice Problem 2.)
od of
bel
lin
28. Let us
all poi
form c
be a li
in P
determ
29. Define
a. Sh
b. Fir
wh
c. W
30. An a
T(x)
that T
transfo
31. Let T
{V1, V2
the set
In Exercise
x = (x1, x2
32. Show
(4x1](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5495aa8f-e580-46c5-af54-23236303b730%2F87e55fe6-f319-4a8f-91cd-8ccd41f49b6e%2F1ovwzl7_processed.jpeg&w=3840&q=75)
Transcribed Image Text:-2
and let
- ()--[3]
V₁ =
-[-3].
5
T: R² → R² be a linear transformation that maps x into
X₁V₁ + X2V₂. Find a matrix A such that T(x) is Ax for
each x.
20. Let x =
X2
and V₂ =
In Exercises 21 and 22, mark each statement True or False. Justify
1 ban
each answer.
21/
21 a. A linear transformation is a special type of function.
b. If A is a 3 x 5 matrix and T is a transformation defined
by T(x) = Ax, then the domain of T is R³.
to 4
lemb1009 1singamos
1091
c.
If A is an m x n matrix, then the range of the transforma-
tion x → Ax is Rm.
d. Every linear transformation is a matrix transformation.
OT DAN DISIqp21
e.
A transformation T is linear if and only if T(C₁V₁+
C2V2) = C₁T(V₁) + C₂T (V₂) for all v₁ and v2 in the
domain of T and for all scalars c₁ and ₂.
22. a. Every matrix transformation is a linear transformation.
b.
The codomain of the transformation x → Ax is the set of
all linear combinations of the columns of A.
n
c. If T: R" → Rm is a linear transformation and if c is
in Rm, then a uniqueness question is "Is c in the range
of T?"
d. A linear transformation preserves the operations of vector
addition and scalar multiplication.
DODENNO
U
linear transformation.
e. The superposition principle is a physical description of a
2
23. Let T: R² → R2 be the linear transformation that reflects
each point through the x₁-axis. (See Practice Problem 2.)
od of
bel
lin
28. Let us
all poi
form c
be a li
in P
determ
29. Define
a. Sh
b. Fir
wh
c. W
30. An a
T(x)
that T
transfo
31. Let T
{V1, V2
the set
In Exercise
x = (x1, x2
32. Show
(4x1
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