# Tagged Questions

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### Finding intersection points of 2 functions. My method is incomplete.

These are the 2 functions : $y = x^{4}-2x^{2}+1$ $y = 1-x^{2}$ Here's how I solved It : $x^{4}-2x^{2}+1 = 1-x^{2}$ $x^{4}-x^{2} = 0$ $x^2(x^2-1)=0$ $x^2-1=0$ $x=\pm \sqrt{1}$ Value of $y$ when ...
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### Linear Algebra Quadratic True False

"Every quadratic form $x^TAx$ with $A$ an invertible matrix is either positive definite, negative definite, or indefinite." Is this true or false? I am just wondering does it have to be positive, ...
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### Does this quadratic form represent 1?

I am stuck on the following question in Lam's quadratic forms for a few days now. Let $a,b,c$ be three elements of a field $F$ such that $0 \neq a^2+b^2 \neq c^2$. Show that the quadratic form ...
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### Let $W \subset V$ vector spaces, let $W'$ be the orthogonal complement of $W$. Show $\dim(W)+\dim(W')=\dim V$

As title says: Let $W \subset V$ vector spaces, let $W'$ be the orthogonal complement of $W$. Show $\dim(W)+\dim(W')=\dim V$. We are given that $W \subset V$ finite vector spaces, symmetric bilinear ...
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### Find the symmetric matrix that represents the quadratic form $Q(X)=trace(X^2)$, $X\in mat_n\mathbb (R)$

as the title says, find the symmetric matrix (or signature) of $Q(X)=trace(X^2)$ where $X$ is an $n$ by $n$ matrix with real entries. the diagonal of $X^2$ is $$\sum_{k=1}^n x_{ik}x_{ki}$$ So ...
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### Signature of quadratic form $Q(p)=p(1)p(2)+p(3)p(4)$

I was asked to find the signature of the quadtratic form $Q(p)=p(1)p(2)+p(3)p(4)$ where $p$ is a polynomial in $\mathbb R_n[x]$ I tried doing it via finding the symmetric matrix that $Q$ corresponds ...