
Matrix rank and nullity
Consider
$\displaystyle A=\[ \left( \begin{array}{ccc}
2 & 8 & 1 \\
3 & 12 & 0 \\
5 & 20 & 2 \end{array} \right)\] $
Consider the linear operator $\displaystyle L_A$ on $\displaystyle R^3$ which is left multiplication by A
What is the rank and nullity of L_A?
Attempt
$\displaystyle L_A (x,y,z)=$ $\displaystyle \[ \left( \begin{array}{ccc} 2x3y+5x & 4(2x3y+5z) & x2z \end{array} \right)\] $
Nullity
So if $\displaystyle x=2z,y=3z, L_A=0$ so the null space of $\displaystyle L_A$ consists of vectors of the form $\displaystyle (2z,3z,z)$ so its basis should be $\displaystyle (2,3,1)$ and thus $\displaystyle nullity(L_A)=1$?
Rank
$\displaystyle {(1,4,0),(0,0,6)}$ should serve as a basis, so $\displaystyle rank(L_A)=2$?

Hmm. Right after the word "Attempt", you have an equation which is incorrect. You're doing pointwise multiplication of the first column times the coordinate vector (almost always treated as column vectors), and then the second column times the coordinate vector, etc. But that's not matrix multiplication. It's rows in the matrix dotted into the coordinate vector. That is,
$\displaystyle L_{A}\begin{bmatrix}x\\y\\z\end{bmatrix}=
\begin{bmatrix}2&8&1\\3&2&0\\5&20&2\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=
\begin{bmatrix}2x+8y+z\\3x2y\\5x+20y2z\end{bmatrix}.
$
However, this operation isn't actually necessary. Just do row reduction on $\displaystyle A$ to find both the rank and the nulllity. What do you get?