The question is: Let T be a linear operator on $V=R^n$ whose matrix $A$ is a real symmetricmatrix.
a) Prove that $V$ is the orthogonal sum $V=(ker T)\oplus(im T)$
b) Prove that $T$ is an orthogonal projection onto $imT$ if and only if, in addition to being symmetric, $A^2=A$.
I can not understand the question very well, $T$ is a linear operation on vector space, then it will operate on some vector $v$ in $V$, then what is kernal of $T$, how can $Tv=1$ in a vector space?