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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. $n$.

If this is not true, is there anything we can do to guarantee a 'high' rank, say bigger than 5? For example by taking $n \gg k$?

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 $n \gg k$?