# Are these systems of equations linear?

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

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 $C$?
$a$ 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
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
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