By the end of this chapter, you should be able to...
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.
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.
Determine if a set of Euclidean vectors spans \(\IR^n\) by solving appropriate vector equations.
Determine if a subset of \(\IR^n\) is a subspace or not.
Determine if a set of Euclidean vectors is linearly dependent or independent by solving an appropriate vector equation.
Explain why a set of Euclidean vectors is or is not a basis of \(\IR^n\text{.}\)
Compute a basis for the subspace spanned by a given set of Euclidean vectors, and determine the dimension of the subspace.
Answer questions about vector spaces of polynomials or matrices.
Find a basis for the solution set of a homogeneous system of equations.
Readiness Assurance.
Before beginning this chapter, you should be able to...
Use set builder notation to describe sets of vectors.