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 = 1c) a T X b = 0 , X ∈ R m × n - yes, v a r = m ∗ n , e q = 1d) A X + X A T = C , X ∈ R m × n - yes, not sure

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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  =  1c)       a    T    X  b  =  0  ,  X  ∈      R          m      ×      n       - yes,   v  a  r  =  m  ∗  n  ,  e  q  =  1d)   A  X  +  X      A    T    =  C  ,  X  ∈      R          m      ×      n       - yes, not sure

Tatiana Gentry · Accepted Answer

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?

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

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