Let V be a vector space of dimension n and let A = {v1, ..., vn} be an ordered basisof V . Suppose w1, ..., wn ∈ V and let (a1j , ..., anj )t denote the coordinates of wjwith respect to A. Put C = [aij ]. Then show that w1, ..., wn is a basis of V if andonly if C is invertible

Let T be a linear transformation from an n dimensional vector space V to an mdimensional vector space W and let C be the matrix of T with respect to a basis Aof V and B of W . Show that(a) rank(T ) = rank(C);

(b) Assume that {v1, v2, · · · , vk} is a basis of Y . Show that the linear transform T : X → Y from (a) is invertible. (Hint: Use (a) to define a suitable linear transform U : Y → X and show that it is the inverse of T .)

We proved a theorem stating that for a linear mapping α : U → V , we havedim Kerα + dim Imα = dimU.A part of the proof went as follows. Assuming dim Kerα = m, there is a basis of Kerα consisting ofm vectors; denote them by u1, . . . , um. Since dim Imα = n, there is a basis of Imα consisting of nvectors; denote them by v1, . . . , vn. Since v1 is in Imα, there is w1 ∈ U such that α(w1) = v1, andsimilarly for all other vectors in the set v1, . . . , vn; in this way, we define n vectors w1, . . . , wn.Now consider a vector u ∈ U . Since α(u) is in Imα, we have α(u) = d1v1 + · · · + dnvn for somenumbers d1, . . . , dn. Consider a vector u′ ∈ U defined as u′ = d1w1 + · · · + dnwn.Prove that u − u′ ∈ Kerα.

We proved a theorem stating that for a linear mapping α : U → V , we havedim Kerα + dim Imα = dimU.A part of the proof went as follows. Assuming dim Kerα = m, there is a basis of Kerα consisting ofm vectors; denote them by u1, . . . , um. Since dim Imα = n, there is a basis of Imα consisting of nvectors; denote them by v1, . . . , vn. Since v1 is in Imα, there is w1 ∈ U such that α(w1) = v1, andsimilarly for all other vectors in the set v1, . . . , vn; in this way, we define n vectors w1, . . . , wn.Now consider a vector u ∈ U . Since α(u) is in Imα, we have α(u) = d1v1 + · · · + dnvn for somenumbers d1, . . . , dn. Consider a vector u′ ∈ U defined as u′ = d1w1 + · · · + dnwn.Continuing this proof, prove that (u − u′) ∈ Kerα.

. If is invertible then is invertible.