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...