Question

A is an n X M matrix. Does (ker a) perpendicular =im (a transpose)?

Answer 1

im trying to understand your question. are you asking if the orthogonal complement of the Ker A is equal to the Im(A^T)?

Answer 2

Exactly. THanks

Answer 3

That is true, thats part of the Fundamental Theorem of Linear Algebra.

Answer 4

i thought that the theorem said that the orthogonal complement of the iamge is the same as the kernel of A^T

they are similar questions, but still different.

I assume you can get from the fundamental theorem to what I asked, I just don't know how

Answer 5

hmm, didnt come out exactly like i wanted <.< those:\[a_i^T\] are supposed to be the rows of the matrix A. Those are also the columns of A^T.

Answer 6

Basically what this proof shows is that any vector in the Ker A is orthogonal to every column of A^T, which means the orthogonal compliment of the Ker A is the Image of A^T.