I have searched a lot about how to write geometric description of span of 3 vectors, but couldn't find anything. There are lot of questions about geometric description of 2 vectors (Span ={v1,V2}) How can I describe 3 vector span?
Question is as follows:
= [−1 2 1] , = [5 0 2] , = [−3 2 2] , = [10 6 9] , = [−6 9 12] ℎ {, , } ℎ {, , }
I have done the first part, please guide me to describe it geometrically?
To find whether some vector $x$ lies in the the span of a set $\{v_1,\cdots,v_n\}$ in some vector space in which you know how all the previous vectors are expressed in terms of some basis, you have to find the solution(s) of the equation $$ a_1 v_1 + \cdots + a_n v_n = x $$ where you have to find all $\{a_1,\cdots,a_n\}$ that satifay the equation. If there are no solutions, then the vector $x$ is not in the span of $\{v_1,\cdots,v_n\}$. If there is at least one solution, then it is in the span.
For the geometric discription, I think you have to check how many vectors of the set = [−1 2 1] , = [5 0 2] , = [−3 2 2] are linearly independent. If there is only one, then the span is a line through the origin. If there are two then it is a plane through the origin. If all are independent, then it is the 3-dimensional space.