I want to show that in $\mathbb{R^4}$ there is an infinite sequence of $2-$dimensional subspaces $W_1, W_2, \dots$ such that for each $i \neq j$ we have $W_i \cap W_j = \{0\}$. How do I go about this? Edit: After some thought I have come up with this. If we write down a 4 by 4 matrix $A = [...

True or False. Let $S=\{v_1, v_2, ...., v_p\}$ be a subset of a vector space $V$. If $\text{span}(S)=V$ then some subset of $S$ is a basis for $V$. My answer: False. Because if $\text{span}(S)=V$ then ALL subsets of $S$ should form a basis for $V$.

I've encountered this situation. Let $p$ be a prime number, $V$ a $\mathbb{Q}$-linear space of dimension $p$ and some basis $v_0,\dots, v_{p-1}$, now let $u_0$ be a vector $$ u_0 = v_{i_1} + \cdots + v_{i_k}, $$ where the indexes are different and some are missing (i.e. $k<p$). Now we are going ...

i m trying to make general case but i cannot find such matrix. For a system of linear equations Ax=b, construct a matrix A and a vector b such that b is not in ColA. i m trying to solve it for 2 days but couldnot understand it.

I joined Mathematics SE community approximately 4 months, and I have not missed two things from the day I joined the community. First is attempting an answer for questions and asking my doubts and second is finding someone in my locality/school/tuition who knew about something so called MATHEMAT...