Abstract Algebra Theory and Applications - Computer Science ...

320 CHAPTER 18 VECTOR SPACES(d) Let {v 1 , . . . , v k } be a basis for the null space of T . We can extend thisbasis to be a basis {v 1 , . . . , v k , v k+1 , . . . , v m } of V . Why? Prove that{T (v k+1 ), . . . , T (v m )} is a basis for the range of T . Conclude that therange of T has dimension m − k.(e) Let dim V = dim W . Show that a linear transformation T : V → W isinjective if and only if it is surjective.16. Let V and W be finite dimensional vector spaces of dimension n over a fieldF . Suppose that T : V → W is a vector space isomorphism. If {v 1 , . . . , v n }is a basis of V , show that {T (v 1 ), . . . , T (v n )} is a basis of W . Conclude thatany vector space over a field F of dimension n is isomorphic to F n .17. Direct Sums. Let U and V be subspaces of a vector space W . The sum ofU and V , denoted U + V , is defined to be the set of all vectors of the formu + v, where u ∈ U and v ∈ V .(a) Prove that U + V and U ∩ V are subspaces of W .(b) If U + V = W and U ∩ V = 0, then W is said to be the direct sumof U and V and we write W = U ⊕ V . Show that every element w ∈ Wcan be written uniquely as w = u + v, where u ∈ U and v ∈ V .(c) Let U be a subspace of dimension k of a vector space W of dimensionn. Prove that there exists a subspace V of dimension n − k such thatW = U ⊕ V . Is the subspace V unique?(d) If U and V are arbitrary subspaces of a vector space W , show thatdim(U + V ) = dim U + dim V − dim(U ∩ V ).18. Dual Spaces. Let V and W be finite dimensional vector spaces over afield F .(a) Show that the set of all linear transformations from V into W , denotedby Hom(V, W ), is a vector space over F , where we define vector additionas follows:(S + T )(v) = S(v) + T (v)(αS)(v) = αS(v),where S, T ∈ Hom(V, W ), α ∈ F , and v ∈ V .(b) Let V be an F -vector space. Define the dual space of V to beV ∗ = Hom(V, F ). Elements in the dual space of V are called linearfunctionals. Let v 1 , . . . , v n be an ordered basis for V . If v =α 1 v 1 +· · ·+α n v n is any vector in V , define a linear functional φ i : V → Fby φ i (v) = α i . Show that the φ i ’s form a basis for V ∗ . This basis iscalled the dual basis of v 1 , . . . , v n (or simply the dual basis if thecontext makes the meaning clear).