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Transformation : Null Space, Range and Eigenvalues

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Please see the attached problem relating to null space, range, eigenvalues and transformation.

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1) For the matrix A3, the null space denoted usually by "ker(A3)" from word "kernel", is defined as the set of solutions of equation:

where (0) is the null vector, that is the vector (0, 0, 0) (unlike number 0, which is a simple scalar)
Trying to solve the above equation, we get the algebraic system:

It can be easily observed that the system is not a Cramer one, because the rank of main matrix is 1, the same as the extended matrix. That means the system is consistent, but an infinite number of solutions will be ...

Solution Summary

A problem relating to null space, range, eigenvalues and transformation is solved. The solution is detailed and well explained.