Is There a General Relationship Between the Range of T and the Null Space of T?

[cocobaby]

Given a linear transformation T from V to V, can we say that the range of T is in the space spanned by the column vectors of T. And we already know that the null space of T is the one spanned by the set of vectors that are orthogonal to the row vectors of T, then is there any general relationship b/t the range of T and the nulll space of T ?

 

[HallsofIvy]

Yes, the "rank-nullity" theorem: If T is a linear transformation from U to V then the nulliity of T (the dimension of the null space of T) plus the rank of T (the dimension of the range of T in V) is equal to the dimension of U.