Linear algebra question

[Darya]

I'm new to linear algebra, trying to understand linear regression, I've encountered an equation

[MATH]C(A)^⟂=N(A^T)[/MATH]
Where A is some matrix, C is its column space, N is a null space.

If I'm not mistaken and the equation is correct, why space orthogonal to column space of A is a null space of A TRANSPOSE? Why to row space of A?

Thanks a lot!!

 

[Singleton]

Hi Darya, the column space is in the target space where as the null space is in the source space. So N(A) wouldn't make sense in the equation.

Look at the dimensions. A: V^m —> W^n
A is nxm. C(A) consists of n-vectors (ie vectors of length n) and they live in W. While N(A) consists of m-vectors from V. But N(A^t) are n-vectors. The equation says more than just the dimensions though.