Is $3x + 4x^2 + y$ a valid linear combination of $x$? I'm not sure if "a linear combination of $x$" allows for other terms than $x$ which would just be treated as constants, or it should strictly contain $x$.
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$\begingroup$A linear combination of one or more variables is a sum of constant multiples of those variables (to the power 1). If there is only one variable, then the only linear combinations of that variable are the constant multiples of that single variable. Thus all linear combinations of the variable $x$ are of the form $px$ for some number $p$.
$\endgroup$ $\begingroup$In short, the answer to your equation is no. Simply because if $x$ is your variable then by definition you cannot treat $x^2$ as a constant, and thus it is a non-linear term. Furthermore, if you decide that $y$ is a constant and not a variable then as shown above $y$ is not of the necessary form $px$ so we will still not have a linear combination.
That said $x + y$ is a valid linear combination of both $x$ and $y$, but you must specify that it is a linear combination of BOTH $x$ and $y$, otherwise it is not.
Hope this helped!
$\endgroup$ $\begingroup$Linear Combination include two operations on a set of vectors i.e.
- Addition of two vector: $x + y$
- Multiplication of scalar to a vector: $c \cdot x$
So $$x = c_1 \cdot x_1 + c_2 \cdot x_2 + \ ...\ + c_n \cdot x_n$$ where $c_i$ is a constant and $x_i$ is any vector (power 1).
$\endgroup$ $\begingroup$A linear combination of $x$ is of the form $ax+bx+cx+...$ where $a,b,c$ are real numbers. So no.
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