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

Banguizb 2022-07-02 Answered
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) A x = b , x R n - yes, classic system of linear equations, v a r = n , e q = m where A R m × n
b) x T A x = 1 , x R n - no, its a quadratic form, v a r = n , e q = 1
c) a T X b = 0 , X R m × n - yes, v a r = m n , e q = 1
d) A X + X A T = C , X R m × n - yes, not sure
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Answers (1)

Tatiana Gentry
Answered 2022-07-03 Author has 10 answers
You need to be careful with (c) and (d). If X, Y M m × n ( R ), and if α, β R , you need to check, for instance, if
a T ( α X + β Y ) b = α ( a T X b ) + β ( a T Y b ) .
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 ( R ), and what is the dimension of the vector space containing C?
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