I have a matrix A= (1 -2 3 -4), (-5 6 -7 8), (9 -10 11 12) ask to find an orthonormal basis for ker(A).
and then verify that V= ( 0 2 4 2) is in ker(A) and express v in terms of the orthonormal basis for ker(A).Pls help me solve tis question, thx.

Question

anonymous · Answer

Start by solving $Ax = 0$ to find the kernel. Then use the Gram-Schmidt process to find a set of orthonormal vectors spanning the kernel.

anonymous · Answer

Do you follow?

anonymous · Answer

yep

anonymous · Answer

how about the second part of the ques?

anonymous · Answer

Once you have an orthonormal basis, expressing any vector in the kernel is as simple as $$x=\sum_{b\in B}{\langle x,b\rangle\over\lVert b\rVert^2} b$$Where $x$ is the vector and $B$ is the basis.

anonymous · Answer

ic ic , thank you very much