I'm working through an exercise in Linear Algebra and its Applications, 4th ed. by D.C. Lay. The question asks:
Let $M_{2×2}$ be the vector space of all $2\times 2$ matrices, and define $T: M_{2×2} \to M_{2×2}$, where $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$. Describe the kernel of the linear transform $T(A)=A+A^T$
Not sure if this helps, but in an earlier part of the question, we discover that the the range of $T$ is the set of $B$ in $M_{2×2}$ with the property that $B^T=B$.
The answer is that the kernel of $T$ is $\{\begin{bmatrix} 0 & b \\ -b & 0 \end{bmatrix}: b \in real\}$
How can we arrive at this answer?
I tried to make $A = \begin{bmatrix} a & b \\ b & a \end{bmatrix}$, then calculate the reduced echelon form of the augmented matrix [$T(A)$ 0]. However, I am not sure if this is the correct approach.
Best Answer
By definition of the kernel you want $T(A)=0$, where $0$ denotes the zero matrix.
So you want $A=-A^T$. Write this out and you'll see this forces $a=-a$, $d=-d$, and $b=-c$.