Find the Kernel

Problem

Ker(S)* where *S*([a],[b],[c])=([a−b−c],[a−b−c],[a−b+c])

Solution

1. Set the output to zero to find the kernel, which consists of all vectors in the domain that map to the zero vector.

S*([a],[b],[c])=([0],[0],[0])

2. Write the system of equations based on the components of the transformation.

a−b−c=0

a−b−c=0

a−b+c=0

3. Simplify the system by observing that the first two equations are identical.

a−b−c=0

a−b+c=0

4. Subtract the equations to solve for c

(a−b+c)−(a−b−c)=0−0

2*c=0

c=0

5. Substitute the value of c back into the first equation to find the relationship between a and b

a−b−0=0

a=b

6. Express the solution vector in terms of a free variable. Let b=t

([a],[b],[c])=([t],[t],[0])=t*([1],[1],[0])

Final Answer

Ker(S)={t*([1],[1],[0]):t∈ℝ}

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