I am asking three different questions on mathematics.?

Question

what is a one to one function?
what is an invertible matrix?
given the vector space of a matrix, how can we construct another vector space which is orthogonal to the former vector space?

Answer 1

A one-to-one function is a function whose values at different points are different. That is if x1=/=x2 then f(x1)=/=f(x2).

There are several equivalent definitions for an invertible matrix. If we go along with the word invertible, it is a matrix whose inverse exists. That is, there is a matrix B such that AB=BA=In where In denotes the n x n identity matrix. Having non-zero determinant is also an often used definition for an invertible matrix.

Now a matrix does not have a vector space. It has a column space, a null space, a row space... So it is a little hard to answer your question right away. But for example the row space and the null space are orthogonal to each other so whatever it is that you are looking for, I believe you need to look up these words first. At the end this will boil down to describing solutions to a system of equations but it seems like you need to read 2-3 sections from a linear algebra book to get there.

Answer 2

1) y=x

2) A square non-singular matirx A has an inverse A^ -1 such that the product of matix A and A^-1 results in the identity matrix

A^-1 = adjoint(A)/ det(A)

3) In matrix theory, a real orthogonal matrix is a square matrix Q whose transpose is its inverse. See link below:
http://en.wikipedia.org/wiki/Orthogonal_matrix

Answer 3

A one-to-one function is a function where each x value has exactly one y value and each y value has exactly one x value. Sorry, I don't know the others.

Answer 4

search in mathworld.org