let $W_1$ be the real vector space of all $5 \times 2$ matrices of such that the sum the sum of the entries in each row is $0 $. let $W_2$be the real vector space of all $5 \times 2$ matrices such that the sum of the entries in each column is zero .find the dimension of the space $W _1 \cap W_2$
my attempts :$\max[0,\dim(W_1)+\dim(W_2)- 5] \le \dim(W_1 ∩ W_2) \le \min[ \dim(W_1) ,\dim (W_2),]$
so $\dim(W_1 ∩ W_2) = 2$
Am i right ?
Elements of $W_1 \cap W_2$ satisfy the condition that the sum of each row and each column is $0$.
Hence the first row completely determines the next row and the first $4$ entries of the first row would determine the $5$ element, hence the dimension is $4$.