Notation for representation of linear function I was reading an algebra book and it talked about linear functions of vectors. It said a linear function of $r$ vectors is a function which is homogeneous and of the first degree in each of the $r$ vectors. A linear function of one vector I think would look like this
$f(X)=a_1x_1+a_2x_2+...+a_nx_n$ if $f$ was a function of the vector $X=(x_1,x_2,...,x_n)$
It then goes on to say for example a linear function of the three vectors
$X=(x_1,x_2,...,x_n)$ $Y=(y_1,y_2,...,y_n)$ $Z=(z_1,z_2,...,z_n)$
 would be of the form
$\sum{a_{ijk}x_iy_jz_k}$
What does this notation actually mean? What is the summation index?
 A: When there is no summation index, you are to assume the most natural choice, in your case, $$\sum a_{ijk}x_iy_jz_k= \sum_{i=1}^n\sum_{j=1}^n
\sum_{k=1}^na_{ijk}x_iy_jz_k.$$
We can also use "words definition" you were given to support this. If we are talking about polynomials in several variables, what it means to be homogeneous of degree $k$ is that $f(\alpha X) = \alpha^k f(X)$, where $X = (x_1,\ldots,x_n)$. What this $k$ tells you is that $f$ is sum of monomials with total degree $k$, for example $f(x,y) = x^2+xy+y^2$ is homogeneous of degree $2$. And you can easily check that $f(\alpha x,\alpha y)= \alpha^2 f(x,y)$.
What you have is that given vectors $X, Y, Z\ldots$ and polynomial $f$, $f$ must be linear (homogeneous of degree $1$) in each vector, i.e. $$f(\alpha X, Y, Z,\ldots) = f(X, \alpha Y, Z,\ldots) = f(X, Y, \alpha Z,\ldots) = \ldots = \alpha f(X, Y, Z,\ldots)$$
What that means that in each monomial of $f$, exactly one $x_i$ appears, exactly one $y_j$ appears and exactly one $z_k$ appears (an etc. if we have more vectors). But there is no limit to the number of monomials or their coefficients. As you can see, this is really described by the sum you have.
