I am given a $w \sim N(0,I_n)$ and $w \in \mathbb{R}^n$ and $X \in \mathbb{R}^{n \times d}$ such that $X_1,..., X_d \in \mathbb{R}^n $ of $X$ that satisfy $\|X_i\|^2 = n$ where $n$ is a scalar and real number. From here I am supposed to derive that $z_i = (w^TX)_i \sim N(0,n)$.
So far I find that $E[(w^TX)_i] = 0$ from mean of $w$.
But I don't understand how I should get the standard deviation $E[(w^TX)_i((w^TX)_i)^T] = n$.
I get that: $$E[(w^TX)_i((w^TX)_i)^T] = E[w^TX_i(w^TX_i)^T] = E[w^TX_iX_i^Tw] $$
From $\|X_i\|^2 = X_iX_i^T = n$ , I get that: $$ E[w^TX_iX_i^Tw] = E[w^Tnw] = nE[w^Tw]$$
From here I don't understand how I should come to the $E[w^Tw] = 1$, and I guess that is something that I should get, but I don't understand how can this be derived and on which basis this works.
Both $w^TX_i$ and $X_i^Tw$ are scalars, so $$w^TX_i X_i^Tw = (w^TX_i)( X_i^Tw) = (X_i^Tw)( w^TX_i)=X_i^Tw w^TX_i$$ and hence $$E[w^TX_i X_i^Tw] = E[X_i^Tw w^TX_i] = X_i^T E[ww^T]X_i = X_i^T I_n X_i^T = X_i^T X_i^T = \|X_i\|^2 = n.$$