MATH 217 Study Guide - Midterm Guide: Orthonormality, Orthonormal Basis, Orthogonal Complement
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If b is a basis of a linear space v, then: [f+g] = [f] +[g, [kf]=k[f] for all f and g in v and all scalars k. If a linear space v has a basis with n elements, then all other bases of v consist of n elements as well. dim (v) = n. If v is finite dimensional, dim v = rank t + nullity t = dim(imt) + dim(kert) If b=(f1, f2, fn) is a basis of v, then the coordinate transformation l (f) = [f] from v to rn is an isomorphism. Thus v is isomorphic to rn; the linear spaces v and rn have the same structure. The columns of the b matrix of a linear transformation. For a linear transformation t from v to v, with b= the matrix t with respect to. The columns of b are the b coordinate vectors of the transformations of the basis elements f1 fn of v.
