Find the Kernel T[[x],[y]]=[[x],[y],[8x+4y]]

Problem

T*[[x],[y]]=[[x],[y],[8*x+4*y]]

Solution

1. Define the kernel by setting the output of the linear transformation equal to the zero vector in the codomain ℝ3

T*[[x],[y]]=[[0],[0],[0]]

2. Set up a system of equations based on the components of the resulting vector.

[[x],[y],[8*x+4*y]]=[[0],[0],[0]]

3. Solve the system by equating the corresponding entries.

x=0

y=0

8*x+4*y=0

4. Verify the consistency of the equations by substituting the values of x and y into the third equation.

8*(0)+4*(0)=0

5. Identify the solution set, which contains only the zero vector in the domain ℝ2

ker(T)={[0],[0]}

Final Answer

ker(T)={[0],[0]}

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