Linear Transformation

Linear Transformation

1. Introduction to Linear Transformation

   • L is one-to-one if for all T(v1) = T(v2) implies v1 = v2
   • The linear transformation
     • 1. T(x, y, z) = (x, y) : Projection
     • 2. T(u) = ru, r > 1 : Dilation
     • 3. T(u) = ru, 0 < r < 1 : Contraction
     • 5. T(u) = : Rotation
     • 4. T(x, y) = (x, -y) : Reflection
     • If T : V -> W is a linear transformation, then for any vectors in V and any scalars, the following properties are true
       • 1.
       • 2. T(-v) = -T(v)
       • 3. T(u - v) = T(u) - T(v)
       • 4.
   • Let V and W be vector spaces. The function T : V -> W, T is called a linear transformation of V into W, following 2 properties
     • 1. T(u+v) =T(u) + T(v), for every u, v element of V
     • 2. T(ku) = kT(u), for every u element of V and every scalar k
     • Linear transformation given by a Matrix, is called matrix transformation

2. The Kernel and Range of A Linear Transformation

   • The kernel of T, denoted by ker(T), is the subset of V consisting of all vectors v such that
     • T : V -> W is a linear transformation, then ker(T) is a subspace of V
   • If T1 : U -> V and T2 : V -> W are LT, then the composition of T2 with T1, denoted by T2 o T1 is a function defined by the formula (T2 o T1) (u bar) = T2(T1(u bar)) where (u bar) is a vector in U
     • T : V -> W is one-to-one iff
   • T is onto iff given any w in W, there is a vector v in V such that T(v) = w
     • If T : V -> W is a LT then T is a subspace of W
     • If T : V -> W is a LT of an n-dimensional vector space V into vector space W, then dim(ker T) + dim(range T) = dimV

3. The Matrix of Linear Transformation

   • Procedure for computing the matrix of a linear transformation
     • 3. The matrix of A of T with respect to S and T is formed by choosing [T(vj)] as j th column of A
     • 2. Express T(vj) as linear combination of the vectors in T
     • 1. Compute T(vj) for j = 1, 2, . . . , n
   • Let T : V -> W be a LT, let S = {vector of v}, T = {vector of w} be bases for V and W respectively. Then m x n matrix A, is associated with T and has the following property
     • If x in V, then

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