T: TR4 R² such that T(1, 0, 0, 0) = (3,-2), T(1, 1, 0, 0) (5, 1), 7(1, 1, 1, 0) = (-1,0), and T(1, 1, 1, 1) = (2, 2). =
T: TR4 R² such that T(1, 0, 0, 0) = (3,-2), T(1, 1, 0, 0) (5, 1), 7(1, 1, 1, 0) = (-1,0), and T(1, 1, 1, 1) = (2, 2). =
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 16EQ
Related questions
Question
can you please solve#28 and show all of your work please, explain each step you make and why is the reason you did it the way.
![For Problems 27-30, assume that T defines a linear trans-
formation and use the given information to find the matrix
of T.
27. T : R² → Rª such that T(−1, 1) = (1, 0, −2, 2) and
T(1, 2) = (–3, 1, 1, 1).
28. TR4 → R² such that T(1, 0, 0, 0) = (3, −2),
T(1, 1, 0, 0) = (5, 1), T(1, 1, 1, 0) = (−1,0), and
T(1, 1, 1, 1) = (2, 2).
29. T : R³ → R³ such that T(1, 2, 0)
=
(2, 1, 1),
T (0, 1, 1) = (3, −1, −1) and T (0, 2, 3) = (6, −5, 4).
30. T R³ R4 such that T(0,−1, 4) = (2, 5, -2, 1),
T (0, 3, 3) (-1, 0, 0, 5), and T(4,4, -1)
(-3, 1, 1, 3).
=
=
31. Let T: P₂ (R) → P₂(R) be the linear transformation
satisfying
T(1) = x+1, T(x)=x²-1, T(x²) = 3x +2.
Determine T (ax² + bx + c), where a, b, and c are
arbitrary real numbers.
32. Let TV → V be a linear transformation, and sup-
pose that
T(2v₁ + 3v₂) = V₁ + V₂,
then T = S; that is, T(v) = S(v) for each v E V.
36. Let V be a vector space with basis {V₁, V2, . . . , Vk} and
suppose TV → W is a linear transformation such
that T (vi) = 0 for each i = 1, 2, ..., k. Prove that T
is the zero transformation; that is, 7 (v) = 0 for each
VE V.
Let T₁ V W and T₂: V → W be linear transforma-
tions, and let c be a scalar. We define the sum T₁+T₂ and
the scalar product cT₁ by
(T₁+T₂) (v) = T₁ (v) + T₂(v)
and
(cT₁)(v) = cT₁ (v)
for all v € V. The remaining problems in this section con-
sider the properties of these mappings.
37. Verify that T₁+T₂ and cT₁ are linear transformations.
38. Let T₁ : R² → R² and T₂ : R² → R² be the linear
transformations with matrices
3 1
2
^-[-2] *-B ]
A =
в
-1
3
9](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0655393b-8df3-4633-b13c-e0d6983d2306%2Faf4e2ade-5493-4c92-a1fe-bed418fdeaea%2Fao584v9_processed.png&w=3840&q=75)
Transcribed Image Text:For Problems 27-30, assume that T defines a linear trans-
formation and use the given information to find the matrix
of T.
27. T : R² → Rª such that T(−1, 1) = (1, 0, −2, 2) and
T(1, 2) = (–3, 1, 1, 1).
28. TR4 → R² such that T(1, 0, 0, 0) = (3, −2),
T(1, 1, 0, 0) = (5, 1), T(1, 1, 1, 0) = (−1,0), and
T(1, 1, 1, 1) = (2, 2).
29. T : R³ → R³ such that T(1, 2, 0)
=
(2, 1, 1),
T (0, 1, 1) = (3, −1, −1) and T (0, 2, 3) = (6, −5, 4).
30. T R³ R4 such that T(0,−1, 4) = (2, 5, -2, 1),
T (0, 3, 3) (-1, 0, 0, 5), and T(4,4, -1)
(-3, 1, 1, 3).
=
=
31. Let T: P₂ (R) → P₂(R) be the linear transformation
satisfying
T(1) = x+1, T(x)=x²-1, T(x²) = 3x +2.
Determine T (ax² + bx + c), where a, b, and c are
arbitrary real numbers.
32. Let TV → V be a linear transformation, and sup-
pose that
T(2v₁ + 3v₂) = V₁ + V₂,
then T = S; that is, T(v) = S(v) for each v E V.
36. Let V be a vector space with basis {V₁, V2, . . . , Vk} and
suppose TV → W is a linear transformation such
that T (vi) = 0 for each i = 1, 2, ..., k. Prove that T
is the zero transformation; that is, 7 (v) = 0 for each
VE V.
Let T₁ V W and T₂: V → W be linear transforma-
tions, and let c be a scalar. We define the sum T₁+T₂ and
the scalar product cT₁ by
(T₁+T₂) (v) = T₁ (v) + T₂(v)
and
(cT₁)(v) = cT₁ (v)
for all v € V. The remaining problems in this section con-
sider the properties of these mappings.
37. Verify that T₁+T₂ and cT₁ are linear transformations.
38. Let T₁ : R² → R² and T₂ : R² → R² be the linear
transformations with matrices
3 1
2
^-[-2] *-B ]
A =
в
-1
3
9
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