EWU Math 231: Vector Spaces - Bases

We define a basis of a vector space V as a linearly independent set that spans V. We see several standard and nonstandard examples of bases. Identifying that several previous theorems were actually giving us a basis of a certain subspace (null space, column space, etc.) we see that columns of a nonsingular matrix are a basis for a vector space of column vectors. We close with a discussion of orthogonality and see how to leverage orthogonality when discussing the unique vector representation of a element of a vector space relative to an orthogonal basis.