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Proving a Subspace is Indeed a Subspace!




These are my notes from Matrices and Vectors MATH 2333 at the University of Texas at Dallas from January 22, 2018. We learn a couple ways to prove a subspace is a subspace.

A subspace of a vector space V is a subset in V, and is itself  a vector space that has underwent the operations of addition and scalar multiplication inherited from the original vector space V. 





In other words the subspace must:
1. Include a zero vector
2. Be closed under addition (I f we add two elements of the subspace is the result in the subspace?)
3. Be closed under scalar multiplication (If we multiply elements in S by some scalar, is the result in S?)
The elements and scalar mentioned will usually belong to all real numbers. 







If any element in a set of vectors is restricted to all positive numbers or all negative numbers (can't be equal to 0) a zero vector is impossible. So always check for the first condition. 





Lastly, all subsets can be expressed as linear combinations or spans. Thus, proving that a set of vectors is a linear span is an alternative way of proving that it is also a subset. 




To do this, take the given set and split it into tuples. If you have a set with two different variables, a and b, and three elements, create two triplets, the first with only one variable, and the second with only the other variable. See notes below. 

This shows that it is closed under addition

Next factor out the variable from each triplet (or tuple) to get the span. 

If you cannot do this, then the set is not a subset, because it is not a linear span. 





If these notes were helpful or need improvement, feel free to comment and share!
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