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 $A\in {R}^{m\times n}$

b) ${x}^{T}Ax=1,x\in {R}^{n}$ - no, its a quadratic form, $var=n,eq=1$

c) ${a}^{T}Xb=0,X\in {R}^{m\times n}$ - yes, $var=m\ast n,eq=1$

d) $AX+X{A}^{T}=C,X\in {R}^{m\times n}$ - yes, not sure

a) $Ax=b,x\in {R}^{n}$ - yes, classic system of linear equations, $var=n,eq=m$ where $A\in {R}^{m\times n}$

b) ${x}^{T}Ax=1,x\in {R}^{n}$ - no, its a quadratic form, $var=n,eq=1$

c) ${a}^{T}Xb=0,X\in {R}^{m\times n}$ - yes, $var=m\ast n,eq=1$

d) $AX+X{A}^{T}=C,X\in {R}^{m\times n}$ - yes, not sure

Tatiana Gentry

Beginner2022-07-03Added 10 answers

You need to be careful with (c) and (d). If $X$, $Y\in {M}_{m\times n}(\mathbb{R})$, and if $\alpha $, $\beta \in \mathbb{R}$, you need to check, for instance, if

${a}^{T}(\alpha X+\beta Y)b=\alpha ({a}^{T}Xb)+\beta ({a}^{T}Yb).$

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 ${M}_{m\times n}(\mathbb{R})$, and what is the dimension of the vector space containing $C$?

${a}^{T}(\alpha X+\beta Y)b=\alpha ({a}^{T}Xb)+\beta ({a}^{T}Yb).$

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 ${M}_{m\times n}(\mathbb{R})$, and what is the dimension of the vector space containing $C$?

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?

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