find the number of solutions of the equation $a_1x_1+a_2x_2+...+a_nx_n=0$ in a linear space over Galois field

Question

Linear space $\Bbb F_p^n$ contains $p^n$ vectors $( x_1, x_2, ..., x_n)$ with length $n$ over finite $\Bbb F_p$ Galois field comprised from $p$ elements.

How many solutions in $\Bbb F_p^n$ has the equation

$\alpha_1x_1+\alpha_2x_2+...+\alpha_nx_n=0$ ( where not all $\alpha_i \in \Bbb F_p$ are zeroes) ?

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This problem is taken from Kostrikin A., I., "Introduction to linear algebra 2".

Answer 1

Hint: the set of solutions forms a vector space over ${\mathbb F}_p$. What is its dimension? What does that tell you about its number of elements?