How to find equation system describing affine space, having base of linear 'overspace' and a vector?
Suppose that I've vectors $\alpha$ and $\beta$, so that $W=\text{lin}(\alpha, \beta)$, and a vector $\theta$. Then $V=W+\theta$. How to describe $V$ using equations system?
I was trying to search this, but coudn't find. Thanks in advance for link or description!
I guess, all these live in some fixed $n$ dimensional space, say in $\Bbb R^n$. If you want to express $W$ by linear equations, you should find orthogonal vectors to $W$: extend the basis of $W$ (mostly it's $(\alpha,\beta)$ in your example) to a basis $(\alpha,\beta,\gamma_1,\gamma_2,..)$ of $\Bbb R^n$ such that all $\gamma_i\perp W$. (This can be done by Gram-Schmidt process, starting out from an arbitrary basis.)
Then we have ${\rm Span}(\gamma_1,\gamma_2,...)^\perp=W$, and an equation system for $W$ can be written by dot product: $$\langle\gamma_1,w\rangle=0 \\ \langle\gamma_2,w\rangle=0 \\ \vdots $$ where $w$ is the variable (vector) and $\gamma_i$ give the coefficients.
Similarly, for the affine subspace $V=W+\Theta$, the equation system is: $$\langle\gamma_1,v\rangle=\langle\gamma_1,\Theta\rangle \\ \langle\gamma_2,v\rangle=\langle\gamma_2,\Theta\rangle \\ \vdots $$