Find the Kernel u=[[1,0],[1,0]]v=[[0,1],[0,1]]

Problem

ker*[[1,0],[1,0]],ker*[[0,1],[0,1]]

Solution

1. Define the kernel for the first matrix A=[[1,0],[1,0]] The kernel consists of all vectors x=[[(x_1)],[(x_2)]] such that A*x=0

2. Set up the system of equations for the first matrix.

[[1,0],[1,0]]*[[(x_1)],[(x_2)]]=[[0],[0]]

3. Solve the system for (x_1) and (x_2)

(x_1)+0*(x_2)=0⇒(x_1)=0

(x_1)+0*(x_2)=0⇒(x_1)=0

The variable (x_2) is free.

4. Express the solution as a span.

[[(x_1)],[(x_2)]]=[[0],[(x_2)]]=(x_2)*[[0],[1]]

5. Define the kernel for the second matrix B=[[0,1],[0,1]] The kernel consists of all vectors y=[[(y_1)],[(y_2)]] such that B*y=0

6. Set up the system of equations for the second matrix.

[[0,1],[0,1]]*[[(y_1)],[(y_2)]]=[[0],[0]]

7. Solve the system for (y_1) and (y_2)

0*(y_1)+(y_2)=0⇒(y_2)=0

0*(y_1)+(y_2)=0⇒(y_2)=0

The variable (y_1) is free.

8. Express the solution as a span.

[[(y_1)],[(y_2)]]=[[(y_1)],[0]]=(y_1)*[[1],[0]]

Final Answer

ker*[[1,0],[1,0]]=span{0},ker*[[0,1],[0,1]]=span{1}

Want more problems? Check here!