  We study delocalization of null vectors and eigenvectors of random matrices
with i.i.d entries. Let $A$ be an $n\times n$ random matrix with i.i.d real
subgaussian entries of zero mean and unit variance. We show that with
probability at least $1-e^{-\log^{2} n}$ $$ \min\limits_{I\subset[n],\,|I|=
m}\|{\bf v}_I\| \geq \frac{m^{3/2}}{n^{3/2}\log^Cn}\|{\bf v}\| $$ for any real
eigenvector ${\bf v}$ and any $m\in[\log^C n,n]$, where ${\bf v}_I$ denotes the
restriction of ${\bf v}$ to $I$. Further, when the entries of $A$ are complex,
with i.i.d real and imaginary parts, we show that with probability at least
$1-e^{-\log^{2} n}$ all eigenvectors of $A$ are delocalized in the sense that
$$ \min\limits_{I\subset[n],\,|I|= m}\|{\bf v}_I\| \geq
\frac{m}{n\log^Cn}\|{\bf v}\| $$ for all $m\in[\log^C{n},n]$.
  Comparing with related results, in the range
$m\in[\log^{C'}{n},n/\log^{C'}{n}]$ in the i.i.d setting and with weaker
probability estimates, our lower bounds on $\|{\bf v}_I\|$ strengthen an
earlier estimate $\min\limits_{|I|= m}\|{\bf v}_I\| \geq c(m/n)^6\|{\bf v}\| $
obtained in [M. Rudelson, R. Vershynin, Geom. Func. Anal., 2016], and bounds
$\min\limits_{|I|= m}\|{\bf v}_I\| \geq c(m/n)^2\|{\bf v}\|$ (in the real
setting) and $\min\limits_{|I|= m}\|{\bf v}_I\| \geq c(m/n)^{3/2}\|{\bf v}\|$
(in the complex setting) established in [K. Luh, S. O'Rourke,
arXiv:1810.00489]. As the case of real and complex Gaussian matrices shows, our
bounds are optimal up to the polylogarithmic multiples. We derive stronger
estimates without the polylogarithmic error multiples for null vectors of real
$(n-1)\times n$ random matrices.
