3 Suppose V is finite-dimensional and TE L(V). Prove that the following are equivalent. (a) T is invertible. n (b) Tv₁,..., Tv is a basis of V for every basis v₁, Un of V. n (c) Tv₁, ..., Tv is a basis of V for some basis v₁, , Un of V.

Complete Example 2 by verifying that {1,x,x2,x3} is an orthonormal basis for P3 with the inner product p,q=a0b0+a1b1+a2b2+a3b3. An Orthonormal basis for P3. In P3, with the inner product p,q=a0b0+a1b1+a2b2+a3b3 The standard basis B={1,x,x2,x3} is orthonormal. The verification of this is left as an exercise See Exercise 17..

In Exercises 7-10, show that the given vectors form an orthogonal basis for2or3. Then use Theorem 5.2 to express was a linear combination of these basis vectors. Give the coordinate vector[w] ofwwith respect to the basis ={v1,v2}of 2or =v1,v2,v3 of3. v1=[111],v2=[110],v3=[112];w=[123]

Determine whether S={1t,2t+3t2,t22t3,2+t3} is a basis for P3.