Question
Let TRR be the linear transformation given by Tz, p.2)=(x+y.2+2) (a) Give a base and dimension of Ker(T); (b) Give a base and dimension of Im(T)
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Answer to a math question Let TRR be the linear transformation given by Tz, p.2)=(x+y.2+2) (a) Give a base and dimension of Ker(T); (b) Give a base and dimension of Im(T)
Rasheed
4.7110 Answers
### Part (a): Kernel of T
Given the linear transformation T(x, y) = (x+y, y+2) T \ker(T)
To find the kernel, we need to solve for x y T(x, y) = 0
T(x, y) = (x+y, y+2) = (0, 0)
This gives us the system of equations:
x + y = 0
y + 2 = 0
Solving these equations, we find:
y = -2
x = 2
Thus, the kernel \ker(T) (2, -2) \ker(T) \{ (2, -2) \} \ker(T)
### Part (b): Image of T
The image of T \text{Im}(T) T(x, y) (x, y) \in \mathbb{R}^2
Expressing any vector in the image as a linear combination of outputs:
(x+y, y+2) = x(1, 0) + y(1, 1) + (0, 2)
The vectors (1, 0) (1, 1)
a(1, 0) + b(1, 1) = (0, 0)
a + b = 0
b = 0
This implies a = 0, b = 0 \text{Im}(T) \{ (1, 0), (1, 1) \} \text{Im}(T)
**Answer:**
- **Kernel of T \{ (2, -2) \}
- **Image of T \{ (1, 0), (1, 1) \}
Given the linear transformation
To find the kernel, we need to solve for
This gives us the system of equations:
Solving these equations, we find:
Thus, the kernel
### Part (b): Image of
The image of
Expressing any vector in the image as a linear combination of outputs:
The vectors
This implies
**Answer:**
- **Kernel of
- **Image of
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