Find the Kernel [[-3,4],[1,2]][[x],[y]]=[[-17],[7]]

Problem

[[−3,4],[1,2]]*[[x],[y]]=[[−17],[7]]

Solution

1. Identify the task. The kernel (or null space) of a matrix A is the set of all vectors v such that A*v=0 The provided equation A*v=b is a non-homogeneous system, but the task asks for the kernel of the transformation matrix.

2. Set up the homogeneous equation. To find the kernel of the matrix A=[[−3,4],[1,2]] we solve the system:

[[−3,4],[1,2]]*[[x],[y]]=[[0],[0]]

3. Write the system as linear equations:

−3*x+4*y=0

x+2*y=0

4. Solve the system using substitution or elimination. From the second equation:

x=−2*y

5. Substitute x=−2*y into the first equation:

−3*(−2*y)+4*y=0

6*y+4*y=0

10*y=0

y=0

6. Determine the value of x

x=−2*(0)=0

7. Conclude that the only solution to the homogeneous system is the zero vector. This means the matrix is non-singular and its kernel contains only the origin.

Final Answer

Kernel*([−3,4],[1,2])={[0],[0]}

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