# Archive | work

## midterm

problems and solutions here.…

## solutions to 2.14

Exercise:
Let $$V$$ be the space of real polynomial functions of degree at most $$3$$. Consider the quadratic form $$q_1:V \to \mathbb{R}$$ given by $$q_1(f):=|f(-1)|^2+|f(0)|^2+|f(1)|^2.$$ Is this form positive definite?
Consider another form $$q:V\to \mathbb{R}$$ given by  …

$$\def\Real{\mathbb{R}}\def\Comp{\mathbb{C}}\def\Rat{\mathbb{Q}}\def\Field{\mathbb{F}}\def\Fun{\mathbf{Fun}}\def\e{\mathbf{e}} \def\f{\mathbf{f}}\def\bv{\mathbf{v}}\def\i{\mathbf{i}} \def\eye{\left(\begin{array}{cc}1&amp;0\\0&amp;1\end{array}\right)} \def\bra#1{\langle #1|}\def\ket#1{|#1\rangle}\def\j{\mathbf{j}}\def\dim{\mathrm{dim}} \def\ker{\mathbf{ker}}\def\im{\mathbf{im}} \def\tr{\mathrm{tr\,}} \def\braket#1#2{\langle #1|#2\rangle}$$
1. Bilinear forms are functions $$Q:U\times U\to k$$ that depend on each of the arguments linearly.