7. Suppose T: R¹0 R¹0 is a linear map. a) If dim (N(T)) = 8, then find dim(R(T)). b) If dim(R(T)) = 10, explain why the transformation is one to one.
7. Suppose T: R¹0 R¹0 is a linear map. a) If dim (N(T)) = 8, then find dim(R(T)). b) If dim(R(T)) = 10, explain why the transformation is one to one.
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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help with question 7
![halt
x 2y
1. Let T : R² → R² be given by T ([:]) = [² +2²]
3y los evitaln
Prove that T is a linear operator using the definition on page 202.
2. Let T: R² →→ R2 be a linear operator. If it is known that
5
2
T ([8]) - [3] and T ([i]) - [1
=
[]
compute T
([
10
2
3. Suppose T: M2x2 → M2x2 and S : M2x2 → M2x2 are linear operators where T(A)
5 1
At + A and S(A) = A - At. For the matrix A =
find
6 0
a) (T+S)(A)
b) (S ○ T)(A)
O
4. Find a basis for the null space of the linear transformation T: R³ R³ given by
(ED)
T
5. Determine if the linear mapping,
is an isomorphism. Explain.
6. Determine if the vector v =
T ( [;)) - [² + ]
x y
5y
=
4
10
-3x +2y + 2z
4x + 5y + z
2y + z
5
T ( [;)) - [ H] [:],
5
1
Y
1)T qam
is in R(T) for T : R² → R² given by
7. Suppose T: R¹⁰ → R¹⁰ is a linear map.
a) If dim(N(T)) = 8, then find dim(R(T)).
b) If dim(R(T)) = 10, explain why the transformation is one to one.
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Transcribed Image Text:halt
x 2y
1. Let T : R² → R² be given by T ([:]) = [² +2²]
3y los evitaln
Prove that T is a linear operator using the definition on page 202.
2. Let T: R² →→ R2 be a linear operator. If it is known that
5
2
T ([8]) - [3] and T ([i]) - [1
=
[]
compute T
([
10
2
3. Suppose T: M2x2 → M2x2 and S : M2x2 → M2x2 are linear operators where T(A)
5 1
At + A and S(A) = A - At. For the matrix A =
find
6 0
a) (T+S)(A)
b) (S ○ T)(A)
O
4. Find a basis for the null space of the linear transformation T: R³ R³ given by
(ED)
T
5. Determine if the linear mapping,
is an isomorphism. Explain.
6. Determine if the vector v =
T ( [;)) - [² + ]
x y
5y
=
4
10
-3x +2y + 2z
4x + 5y + z
2y + z
5
T ( [;)) - [ H] [:],
5
1
Y
1)T qam
is in R(T) for T : R² → R² given by
7. Suppose T: R¹⁰ → R¹⁰ is a linear map.
a) If dim(N(T)) = 8, then find dim(R(T)).
b) If dim(R(T)) = 10, explain why the transformation is one to one.
=
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