# Binomial representation of vectors

In my notes I have the following representation:

$$(a+\beta'X)^2$$

where $$a$$ is a constant and both $$\beta$$ and $$X$$ are k by 1 vectors.

The representation is further written as follows:

$$a^2+2a\beta'X+X'\beta \beta'X)$$

While the first part is clear to me, I struggle to understand why it is shown once with $$'$$ ($$X'$$ and $$\beta'$$) and once without $$'$$ ($$X$$ and $$\beta$$).

$$'$$ here denotes the transpose vector. Since both $$\beta$$ and $$X$$ are $$k\times 1$$ vectors, $$\beta' X$$ is a scalar, so $$\beta'X = (\beta'X)' = X'\beta$$, and $$(\beta'X)^2 = (\beta'X)(\beta'X) = X'\beta\beta'X.$$