Finding the nullity of a linear transformation on 3×3 matrices

[Punkyc7]

Let M be a 3x3 matrix with entries from the Reals

Let T:V-W be linear


such that

T(M)= M+M^t


what is the nullity


we know

dimN(T)+dimR(T)=Dim V


im not sure how would figure this out, if the number were abritary numbers couldn't you have a nullity of 9 to 0?

 

[spamiam]

The nullity is the dimension of the null space. If M is in the null space of T, what does T(M) equal?

 

[HallsofIvy]

You are given that T maps M to M+ M^t.

That is, T maps
$$\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}$$
to
$$\begin{bmatrix}2a & b+ d & c+ g \\ b+ d & 2e & f+ h \\ c+ g & f+h & 2i\end{bmatrix}$$

So any matrix in the nullspace of T must satisfy 2a= 0, b+ d= 0, c+ g= 0, 2e= 0, c+ g= 0, 2i= 0. The dimension of the space of all 3 by 3 matrices is 9, of course, but now we are imposing 6 conditions on it.

(In fact, the nullspace of T is exactly the space of all anti-symmetric 3 by 3 matrices.)