Geometric description of span of 3 vectors

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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:

𝑙𝑒𝑡 𝒗𝟏 = [−1 2 1] , 𝒗𝟐 = [5 0 2] , 𝒗𝟑 = [−3 2 2] , 𝒃𝟏 = [10 6 9] , 𝒃𝟐 = [−6 9 12] 𝐷𝑒𝑡𝑒𝑟𝑚𝑖𝑛𝑒 𝑖𝑓 𝒃𝟏 𝑎𝑛𝑑 𝒃𝟐 𝑙𝑖𝑒𝑠 𝑖𝑛 𝑡ℎ𝑒 𝑠𝑝𝑎𝑛{𝒗𝟏, 𝒗𝟐, 𝒗𝟑} 𝑎𝑛𝑑 𝑎𝑙𝑠𝑜 𝑒𝑥𝑝𝑙𝑎𝑖𝑛 𝑡ℎ𝑒 𝑔𝑒𝑜𝑚𝑒𝑡𝑟𝑖𝑐 𝑑𝑒𝑠𝑐𝑟𝑖𝑝𝑡𝑖𝑜𝑛 𝑜𝑓 𝑠𝑝𝑎𝑛{𝒗𝟏, 𝒗𝟐, 𝒗𝟑}

I have done the first part, please guide me to describe it geometrically?

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1 Answer

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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 𝒗𝟏 = [−1 2 1] , 𝒗𝟐 = [5 0 2] , 𝒗𝟑 = [−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.

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