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### Visualize and define a vector space without dot / inner product

I'm trying to rebase my know how in linear algebra, restart from scratch to get a more formal and useful set of definitions to help me with computer programming stuff . One of the first concepts is a ...
I have the following three equations: $$\cos\theta \left(\cos\psi - k_3\sin\psi\right) = k_1$$ $$\sin\phi\sin\theta\cos\psi - \cos\phi\sin\psi - k_3\left(\cos\phi\cos\psi + \sin\phi\sin\theta\sin\... 2answers 539 views ### Solution to a system of symmetric equations After applying the Lagrange multiplier method, I got the following system of equations, which is quite symmetric: (x+y)^2 + (x+z)^2 = \frac{2}{3} \lambda x (y+x)^2 + (y+z)^2 = \frac{2}{3} \lambda ... 4answers 89 views ### Solving a set of 3 Nonlinear Equations In the following 3 equations:$$ k_1\cos^2(\theta)+k_2\sin^2(\theta) = c_1  2(k_2-k_1)\cos(\theta)\sin(\theta)=c_2  k_1\sin^2(\theta)+k_2\cos^2(\theta) = c_3 $$c_1, c_2 and c_3 are ... 2answers 661 views ### Checking if a System of Polynomial Equations is Consistent I'm trying to determine whether any solutions exist to a system of (n+1) polynomial equations in n unknowns. For example:$$ \begin{align*} xy&=-2\\ x^2-1&=0\\ y^3-3y^2+2y&=0 \end{...
In high energy physics, one often encounters conservation of momentum and energy equations of the following form: \begin{array}{rcr} \sum_i (-1)^{\alpha_i}\sqrt{k_i^2 + m_i^2} = 0 \\ \sum_i (-1)^{\...