Find equations for $W$. and Find the $dim(W)$

Question

Let $W$ the subspaces of $Mat_n(\mathbb{R})$ formed by all matrix such that the sum of their rows is equal to zero.

$W=\{A\in Mat_n\mathbb({R}):a_{11}+...+a_{1n}=...=a_{n1}+...+a_{nn}=0\}$

a)Find equations for $W$.
b)Find the $dim(W)$

For other question the buddy @Gribouillis help me to find a basis for $W^\perp$.

His proof

Let $\varphi_1,\ldots,\varphi_n$ be the linear forms sum of the $k$-th row $$\varphi_k(A) = a_{k,1}+\cdots+a_{k, n}$$ Obviously, $A\in W \Rightarrow \varphi_k(A) = 0$, hence $\varphi_k\in W^\bot$. Let $S = \text{Span}(\varphi_1,\ldots,\varphi_n)\subset W^\bot$, then $A\in S^\bot \Longleftrightarrow \forall k\in [1,n], \sum_j a_{k,j}=0 \Longleftrightarrow A\in W$. Hence $W^\bot = (S^\bot)^\bot = S$ and $\varphi_1,\ldots,\varphi_n$ is a basis of $W^\bot$.

Edit: Let us recall the definition of $S^\bot = \{A \in Mat_n(\mathbb{R})\ | \ \forall \alpha \in S, \alpha(A) = 0\}$.

Edit2: Remark that the forms $\varphi_1,\ldots,\varphi_k$ are independent because if for $\lambda_i\in\mathbb{R}$ $$\alpha := \lambda_1 \varphi_1+\cdots+\lambda_n\varphi_n = 0,$$ then one has $0 = \alpha(E_{k,k}) = \lambda_k$ where $E_{k,k}$ is the matrix which all terms are $0$ but the $k$-th term on the diagonal which value is $1$.

My work:

b) We know $dim(W)+dim(W^\perp)=dim(Mat_n(\mathbb{R}))\Rightarrow dim(W)=dim(Mat_n(\mathbb{R}))-dim(W^\perp)=n^2-n$.

for the part a) i have a doubt.

We know $W=(W^\perp)^\perp\rightarrow W=\{A\in Mat_{n}(\mathbb{R})|f(A)=0\forall f\in W^{\perp}\}=\left\{A\in Mat_{n}(\mathbb{R})\,\middle|\,\begin{cases} f_{1}(A)=0\\ .\\ .\\ .\\ f_{n}(A)=0 \end{cases}\right\}$

Then a system of equations for $W$ is

$\begin{cases} a_{k1}+...+a_{kn}=0\\ .\\ .\\ . \end{cases}$

Is correct this? I have a serious doubt about this last part. Thanks for help, and for your time!

Answer 1

The last part will be correct if $f_k$ is replaced by $\varphi_k$. However, note that the equations that you get are the equations that you started with, in the definition of $W$ (in spite of the fact that you only called them $\varphi_k$ later).

Answer 2

A simple equation for $W$ is $$ W=\{M\in\operatorname{Mat}_n(\mathbb{R}):Mu=0\} $$ where $u=[1\ 1\ \dotsm\ 1]^T$ is the $n\times1$ column with all $1$ entries.

In particular, $W=\ker T$, where $T$ is the linear map $$ T\colon\operatorname{Mat}_n(\mathbb{R})\to\mathbb{R}^n \qquad T(M)=Mu $$ This map is easily seen to be surjective.

By the rank-nullity theorem, $$ \dim W=\dim\ker T=\dim\operatorname{Mat}_n(\mathbb{R})-\dim\mathbb{R}^n=n^2-n $$