Banguizb

2022-07-02

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

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 $C$?

Do you have a similar question?