Let
F
F
be a nontrivial quadratic form in
N
N
variables with coefficients in a number field
k
k
and let
Z
\mathcal {Z}
be a subspace of
k
N
{k^N}
of dimension
M
,
1
≤
M
≤
N
M,1 \leq M \leq N
. If
F
F
restricted to
Z
\mathcal {Z}
vanishes on a subspace of dimension
L
,
1
≤
L
>
M
L,1 \leq L > M
, and if the rank of
F
F
restricted to
Z
\mathcal {Z}
is greater than
M
−
L
M - L
, then we show that
F
F
must vanish on
M
−
L
+
1
M - L + 1
distinct subspaces
X
0
,
X
1
,
…
,
X
M
−
L
{\mathcal {X}_0},{\mathcal {X}_1}, \ldots ,{\mathcal {X}_{M - L}}
in
Z
\mathcal {Z}
each of which has dimension
L
L
. Moreover, we show that for each pair
X
0
,
X
1
,
1
≤
l
≤
M
−
L
{\mathcal {X}_0},{\mathcal {X}_1},1 \leq l \leq M - L
, the product of their heights
H
(
X
0
)
H
(
X
1
)
H({\mathcal {X}_0})H({\mathcal {X}_1})
is relatively small. Our results generalize recent work of Schlickewei and Schmidt.