V1: I can explain why a given set with defined addition and scalar multiplication does satisfy a given vector space property, but nonetheless isn't a vector space.
V2: I can determine if a Euclidean vector can be written as a linear combination of a given set of Euclidean vectors by solving an appropriate vector equation.
V3: I can determine if a set of Euclidean vectors spans \(\IR^n\) by solving appropriate vector equations.
V4: I can determine if a subset of \(\IR^n\) is a subspace or not.
V5: I can determine if a set of Euclidean vectors is linearly dependent or independent by solving an appropriate vector equation.
V6: I can explain why a set of Euclidean vectors is or is not a basis of \(\IR^n\text{.}\)
V7: I can compute a basis for the subspace spanned by a given set of Euclidean vectors, and determine the dimension of the subspace.
V8: I can answer questions about vector spaces of polynomials or matrices.
V9: I can find a basis for the solution set of a homogeneous system of equations.