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LINEAR TRANSFORMATIONS ROHAN RAMCHAND Let V, W be two vector spaces over a field F . Definition 1. A linear transformation is a mapping T : V → W such that for u, v ∈ V and λ ∈ F , T (u + v) = T (u) + T (v) T (0) = 0 T (−u) = −T (u) T (λu) = λT (u) Let T : F n → F m be a linear transformation from an n-dimensional field to itself. Then T can be represented as an m-dimensional vector of linear transformations from F n to F: T = (T1 , T2 , ..., Tm ), Ti : F n → F. Further, each Ti is a linear transformation: Ti (x1 , x2 , ..., xn ) = ai1 x1 + ai2 x2 + ... + ain xn . Then any linear transformation can be represented as an n × m matrix: T = a11 a21 .. . a12 a22 .. . a1n a2n .. . ... ... .. . am am . . . am n 1 2 Example. Let T (x, y) = (x + y, x − y) be a linear transformation from R2 → R2 . Then T = 1 1 1 −1 . Example. Let X, Y be finite sets and ϕ : X → Y be a map from X to Y . Let F (X), F (Y ) be the field of F -valued functions on X and Y , respectively, and let ϕ∗ : F (Y ) → F (X) be a mapping of fields such that ϕ∗ (f ) = f ◦ ϕ. Then ϕ∗ is a linear transformation. 1 LINEAR TRANSFORMATIONS 2 Proof. ϕ∗ (f + g)(x) = (f + g)(ϕ(x)) = f (ϕ(x)) + g(ϕ(x)) = ϕ∗ (f )(x) + ϕ∗ (g)(x) ϕ∗ (λf ) = (λf )(ϕ(x)) = λf (ϕ(x)) = λϕ∗ (f )(x) Therefore, ϕ∗ satisfies both properties of a linear transformation and the proof is complete. Operations on Linear Transformations We define two operations on linear transformations, addition and multiplication by a scalar. Definition 2. Let T, S : V → W be linear transformations from a vector space V to a vector space W over the field F . Then (T + S)(v) = T (v) + S(v). Definition 3. Let T : V → W be a linear transformation from a vector space V to a vector space W over the field F . Then for some λ ∈ F , (λ · T )(v) = λ · T (v). The following theorem is presented without proof. Theorem 1. Let T, S : V → W and λ ∈ F as above. Then both T + S and λ · T are linear transformations. Definition 4. Let V and W be two vector spaces over a field F . Then Hom(V, W ) is the set of linear transformations from V to W . Theorem 2. Let + : Hom(V, W ) × Hom(V, W ) → Hom(V, W ) and α : F × Hom(V, W ) → Hom(V, W ) the operations on linear transformations as defined above. Then the triple (Hom(V, W ), +, α) is a vector space over F . (Hom(V, W ) is also referred to as the tensor product of V and W -dual, V L W + .) Proof. (Hom(V, W ), +) is an abelian group by definition. Let 0 : V → W be the identity function on Hom(V, W ); then 0V = 0W ∀ v ∈ V . Finally, define the additive inverse as (−T )(v) = T (−v) = −T (v). Then (Hom(V, W ), +, α) is a vector space. LINEAR TRANSFORMATIONS 3 Note that Hom(Hom(V, W ), V ) is a vector space, as is Hom(Hom(Hom(V, W ), V ), W ), etc. This is known as the self-reducibility of linear algebra. 1 a1 . . . a1n Let T, S : F n → F m be linear transformations. Then, as defined above, T ↔ ... . . . ... am . . . am n 1 1 b1 . . . b1n c11 . . . c1n and S ↔ ... . . . ... . Then T + S ↔ ... . . . ... , where cji = aji + bji . bm . . . bm cm . . . cm n n 1 1 Definition 5. Let Matm×n (F ) be the set of all m × n matrices with elements in F . Let + : Mat(F ) × Mat(F ) → Mat(F ) and α : F × Mat(F ) → Mat(F ); (Matm×n (F ), +, α) is a vector space. With this definition, the following theorem is presented without proof. Theorem 3. Let Hom(F n , F m ) be the set of linear transformations from F n to F m , and let Matm×n (F ) be the set of m × n matrices with elements in F . Then Hom(F n , F m ) ∼ = Matm×n (F ). We now define one final operation on linear transformations, composition of transformations. Definition 6. Let U, V, W be three vector spaces over a field F . Let T : U → V and S : V → W be linear transformations. Then we define composition of linear transformations as S ◦ T : U → W = S(T (u)), u ∈ U . Theorem 4. Let S ◦ T be the composition of linear transformations S and T . Then S ◦ T is a linear transformation. Proof. (S ◦ T )(u1 + u2 ) = S(T (u1 + u2 )) = S(T (u1 ) + T (u2 )) = S(T (u1 )) + S(T (u2 )) (S ◦ T )(λ · u) = S(T (λ · u)) = S(λ · T (u)) = λ · S(T (u)) Then S ◦ T is a linear transformation and the proof is complete. LINEAR TRANSFORMATIONS 4 n n Finally, as1 above, 1letT, S : F → F 1 be linear transformations. Then, 1 as defined above, a1 . . . a n b1 . . . b1n c1 . . . c1n T ↔ ... . . . ... and S ↔ ... . . . ... . Then S ◦ T ↔ ... . . . ... , where an1 . . . ann bn1 . . . bnn cn1 . . . cnn cji = b1i aj1 +b2i aj2 +...+bni ajn . Therefore, S ◦T = T S. (This is presented without proof.)