ag.algebraic geometry - Rank of a linear combination of quadratic forms

Suppose we have a set of quadratic forms $Q_i (x_1, dots, x_n)$ for $1 leq i leq k$ in $n$ variables, defined over $mathbb{R}$. We suppose these are 'collectively nondegenerate' in the sense that there does not exist a change of variables which takes us into a set of quadratic forms with less than $n$ variables.

I am looking at linear combinations of these forms: $$Q_{boldsymbol{lambda}}(textbf{x})=sum_i lambda_i Q_i(x_1, dots, x_n)$$ for $boldsymbol{lambda} = (lambda_1, dots , lambda_k) in mathbb{R}^k$. My question is whether we are guaranteed a set of $lambda$s which gives us a quadratic form of full rank i.e. $n$? Edit:: this has been shown to be untrue, so...

Is there anything we can do to guarantee a 'high' rank, say bigger than 5? For example by taking $n gg k$?

• Next ag.algebraic geometry - Relation between characteristic variety and support of D-Module
• Previous enumerative geometry - Chow Ring of Moduli Space of Abelian Varieties