Final answer:
To find the kernel of a linear transformation, identify the knowns, solve the appropriate equations for the unknowns, substitute known values, and check if the solutions are reasonable.
Step-by-step explanation:
To find the kernel of a linear transformation, one must first understand that the kernel consists of all vectors that are mapped to the zero vector by that transformation. The process of finding the kernel includes a few important steps:
- Make a list of what is given or can be inferred from the problem as stated (identify the knowns).
- Solve the appropriate equation or equations for the quantity to be determined (the unknown). It can be useful to think in terms of a translational analog because by now you are familiar with such motion.
- Substitute the known values along with their units into the appropriate equation, and obtain numerical solutions complete with units. Be sure to use units of radians for angles, if relevant.
- Check your answer to see if it is reasonable: Does your answer make sense?
For example, if the linear transformation is given in terms of a matrix, to find the kernel, set the matrix equation A•x = 0 and solve for the vector x that satisfies this equation. This often requires performing Gaussian elimination or applying other methods of solving systems of linear equations.