Let $t\in \mathbb{R}$ and the vectors $$v_1=\begin{pmatrix}0\\ 1\\ -1\\ 1\end{pmatrix}, v_2=\begin{pmatrix}t\\ 2\\ 0\\ 1\end{pmatrix}, v_3=\begin{pmatrix}2\\ 2\\ 2\\ 0\end{pmatrix}$$ in $\mathbb{R}^4$.

I want to determine a maximal linearly independent subset of $\{v_1, v_2, v_3\}$ and to extend these to a basis of $\mathbb{R}^4$.

I have shown the following:

If $t\neq 1$ the $3$ vectors are linearly independent, so the linearly independent subset is the whole set.

If $t=1$ we get for example $v_3=-2v_1+2v_2$ and $v_1$ and $v_2$ are linearly independent, so a linearly independent subset…

[Division by zero] Caption:
1. The mere existence of zero divisors in associative algebras places heavy restrictions on the zero terms. When these were given inverses, further restrictions were imposed by associativity.
2. There are a total of 7 distinct cases for each sided multiplicative identity (and the two sided identity cases are given by taking the intersection of the suitable pairs of the total of 14 case). Of the 7 base cases, if any product xy equals a simple cube e.g. zzz, then xy=yx for the given x and y involved. One consequence is that at least one zero term commutes if a nilp…