Are these sets of equations linear? What is the number of variables and equations in each system? Pl

Are these sets of equations linear? What is the number of variables and equations in each system? Please correct me if my answer is wrong:
a) $Ax=b,x\in {R}^{n}$ - yes, classic system of linear equations, $var=n,eq=m$ where $A\in {R}^{m×n}$
b) ${x}^{T}Ax=1,x\in {R}^{n}$ - no, its a quadratic form, $var=n,eq=1$
c) ${a}^{T}Xb=0,X\in {R}^{m×n}$ - yes, $var=m\ast n,eq=1$
d) $AX+X{A}^{T}=C,X\in {R}^{m×n}$ - yes, not sure
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Tatiana Gentry
You need to be careful with (c) and (d). If $X$, $Y\in {M}_{m×n}\left(\mathbb{R}\right)$, and if $\alpha$, $\beta \in \mathbb{R}$, you need to check, for instance, if
${a}^{T}\left(\alpha X+\beta Y\right)b=\alpha \left({a}^{T}Xb\right)+\beta \left({a}^{T}Yb\right).$
As for the the number of variables and equations, the number of variables is the dimension of the vector space containing your unknown quantity $x$ or $X$, and the number of equations is the dimension of the vector space where your equation exists. For example, in (d), what is the dimension of ${M}_{m×n}\left(\mathbb{R}\right)$, and what is the dimension of the vector space containing $C$?