7. T: R³ R² defined by T(x) = Ax, where 1 3 2 265 A =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
SOLVE NUMBER 7 BY SHOWING ALL OF YOUR STEPS AND POST PICTURES OF YOUR WORK PLEASE, EXPLAIN EACH STEP YOU MAKE
![Differential Equations and Linear Algebra.pdf
Page 414 of 871
406
True-False Review
For Questions (a)-(f), decide if the given statement is true
or false, and give a brief justification for your answer. If true,
you can quote a relevant definition or theorem from the text.
If false, provide an example, illustration, or brief explanation
of why the statement is false.
(a) If T V → W is a linear transformation and W is
finite-dimensional, then
dim[Ker(T)] + dim[Rng(7)] = dim[W].
(b) If T: P4 (R) → R7 is a linear transformation, then
Ker(7) must be at least two-dimensional.
(c) If T R → Rm is a linear transformation with matrix
A, then Rng (T) is the solution set to the homogeneous
linear system Ax 0.
(d) The range of a linear transformation T: V → W is a
subspace of V.
(e) If T: M23 (R) → P7(R) is a linear transformation
1 1
123
= 0, then
000
4 5 6
Rng (T) is at most four-dimensional.
with T
1]
=
CHAPTER 6
= 0 and T
Linear Transformations
7. T : R³ → R² defined by T (x) = Ax, where
1 3 2
265
A =
(C) X = (35, 25, −5).
For Problems 3-7, find Ker(T) and Rng(T), and give a ge-
ometrical description of each. Also, find dim[Ker(7)] and
dim[Rng(7)], and verify Theorem 6.3.8.
3. T: R². → R² defined by T (x)
6
A-[139]
=
Search
4. T : R³ → R³3 defined by T(x) =
=
A = 0
A =
=
1 10
12
2 −1 1
6. T : R³ → R² defined by T(x)
A = [_3
5. T : R³ → R³ defined by T(x) = Ax, where
1
-2 1
2
-3 -1
5 -8 −1
Ax, where
Ax, where
= Ax, where
1 −1 2
-3 3 -6
(a) Show that Ker(7) consists of all polynomials of
the form b(x - 2), and hence, find its dimension.
(b) Find Rng(7) and its dimension.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0655393b-8df3-4633-b13c-e0d6983d2306%2Fa18ca1d4-8e03-4c2e-81a1-e01a0afb08cc%2Fszntthl_processed.png&w=3840&q=75)
Transcribed Image Text:Differential Equations and Linear Algebra.pdf
Page 414 of 871
406
True-False Review
For Questions (a)-(f), decide if the given statement is true
or false, and give a brief justification for your answer. If true,
you can quote a relevant definition or theorem from the text.
If false, provide an example, illustration, or brief explanation
of why the statement is false.
(a) If T V → W is a linear transformation and W is
finite-dimensional, then
dim[Ker(T)] + dim[Rng(7)] = dim[W].
(b) If T: P4 (R) → R7 is a linear transformation, then
Ker(7) must be at least two-dimensional.
(c) If T R → Rm is a linear transformation with matrix
A, then Rng (T) is the solution set to the homogeneous
linear system Ax 0.
(d) The range of a linear transformation T: V → W is a
subspace of V.
(e) If T: M23 (R) → P7(R) is a linear transformation
1 1
123
= 0, then
000
4 5 6
Rng (T) is at most four-dimensional.
with T
1]
=
CHAPTER 6
= 0 and T
Linear Transformations
7. T : R³ → R² defined by T (x) = Ax, where
1 3 2
265
A =
(C) X = (35, 25, −5).
For Problems 3-7, find Ker(T) and Rng(T), and give a ge-
ometrical description of each. Also, find dim[Ker(7)] and
dim[Rng(7)], and verify Theorem 6.3.8.
3. T: R². → R² defined by T (x)
6
A-[139]
=
Search
4. T : R³ → R³3 defined by T(x) =
=
A = 0
A =
=
1 10
12
2 −1 1
6. T : R³ → R² defined by T(x)
A = [_3
5. T : R³ → R³ defined by T(x) = Ax, where
1
-2 1
2
-3 -1
5 -8 −1
Ax, where
Ax, where
= Ax, where
1 −1 2
-3 3 -6
(a) Show that Ker(7) consists of all polynomials of
the form b(x - 2), and hence, find its dimension.
(b) Find Rng(7) and its dimension.
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