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

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

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

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

Thanks for any help..

EDIT: are the first 3 solutions correct now?

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Be careful about c); it is different than b) in a very critical way. –  Eric Stucky Jan 3 '13 at 9:35
    
Check the # of eqns you have for (b) too. –  anon Jan 3 '13 at 9:39
    
$var = n$ , this is true, right? Since $x \in R^n$, and the number of equations must be somehow proportional to the size of $A$... is it $eq = m*n$? –  Smajl Jan 3 '13 at 9:49
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1 Answer 1

up vote 3 down vote accepted

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 $$a^T(\alpha X + \beta Y) b = \alpha (a^T X b) + \beta (a^T Y b).$$

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

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$a$ is a vector and $b$ too –  Smajl Jan 3 '13 at 9:46
    
So, what sized matrix is the product $a^T X b$, if $a \in \mathbb{R}^m$, $X \in M_{m \times n}(\mathbb{R})$, and $b \in \mathbb{R}^n$? –  Branimir Ćaćić Jan 3 '13 at 9:48
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well, its a scalar ($R^{1 \times 1}$) made of all the products so $eq = 1$ and $var = n$? –  Smajl Jan 3 '13 at 9:54
    
Almost. What is the dimension of the vector space $M_{m \times n}(\mathbb{R}) = \mathbb{R}^{m \times n}$ of all $m \times n$ real matrices? For this is the vector space containing your unknown $X$ in this case. –  Branimir Ćaćić Jan 3 '13 at 10:07
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Size of $C$ is $m*m$ so thats should be the number of equations and number of variables is $m*m$ too... –  Smajl Jan 3 '13 at 10:30
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