Linear Transformations-Linear Algebra-Lecture 19 Slides-Mathematics, Slides of Linear Algebra

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MATH 304

Linear Algebra Lecture 19: Linear transformations. Kernel and range.

Linear mapping = linear transformation = linear function

Definition. Given vector spaces V 1 and V 2 , a mapping L : V 1 → V 2 is linear if L(x + y) = L(x) + L(y), L(r x) = rL(x) for any x, y ∈ V 1 and r ∈ R.

A linear mapping ℓ : V → R is called a linear functional on V. If V 1 = V 2 (or if both V 1 and V 2 are functional spaces) then a linear mapping L : V 1 → V 2 is called a linear operator.

Properties of linear mappings

Let L : V 1 → V 2 be a linear mapping.

• L(r 1 v 1 + · · · + rk vk ) = r 1 L(v 1 ) + · · · + rk L(vk ) for all k ≥ 1, v 1 ,... , vk ∈ V 1 , and r 1 ,... , rk ∈ R. L(r 1 v 1 + r 2 v 2 ) = L(r 1 v 1 ) + L(r 2 v 2 ) = r 1 L(v 1 ) + r 2 L(v 2 ), L(r 1 v 1 + r 2 v 2 + r 3 v 3 ) = L(r 1 v 1 + r 2 v 2 ) + L(r 3 v 3 ) = = r 1 L(v 1 ) + r 2 L(v 2 ) + r 3 L(v 3 ), and so on.
• L( 01 ) = 02 , where 01 and 02 are zero vectors in V 1 and V 2 , respectively. L( 01 ) = L(0 01 ) = 0L( 01 ) = 02.
• L(−v) = −L(v) for any v ∈ V 1. L(−v) = L((−1)v) = (−1)L(v) = −L(v).

Examples of linear mappings

• Scaling L : V → V , L(v) = sv, where s ∈ R. L(x + y) = s(x + y) = sx + sy = L(x) + L(y), L(r x) = s(r x) = r (sx) = rL(x).
• Dot product with a fixed vector ℓ : Rn^ → R, ℓ(v) = v · v 0 , where v 0 ∈ Rn. ℓ(x + y) = (x + y) · v 0 = x · v 0 + y · v 0 = ℓ(x) + ℓ(y), ℓ(r x) = (r x) · v 0 = r (x · v 0 ) = r ℓ(x).
• Cross product with a fixed vector L : R^3 → R^3 , L(v) = v × v 0 , where v 0 ∈ R^3.
• Multiplication by a fixed matrix L : Rn^ → Rm, L(v) = Av, where A is an m×n matrix and all vectors are column vectors.

Linear differential operators

• an ordinary differential operator

L : C ∞(R) → C ∞(R), L = g 0 d^2 dx^2

• g 1 d dx

• g 2 ,

where g 0 , g 1 , g 2 are smooth functions on R. That is, L(f ) = g 0 f ′′^ + g 1 f ′^ + g 2 f.

• Laplace’s operator ∆ : C ∞(R^2 ) → C ∞(R^2 ),

∆f =

∂^2 f ∂x^2

∂^2 f ∂y 2 (a.k.a. the Laplacian; also denoted by ∇^2 ).

Range and kernel

Let V , W be vector spaces and L : V → W be a linear mapping. Definition. The range (or image) of L is the set of all vectors w ∈ W such that w = L(v) for some v ∈ V. The range of L is denoted L(V ). The kernel of L, denoted ker L, is the set of all vectors v ∈ V such that L(v) = 0.

Theorem (i) The range of L is a subspace of W. (ii) The kernel of L is a subspace of V.

Example. L : R^3 → R^3 , L

x y z

x y z

The range of L is spanned by vectors (1, 1 , 1), (0, 2 , 0), and (− 1 , − 1 , −1). It follows that L(R^3 ) is the plane spanned by (1, 1 , 1) and (0, 1 , 0). To find ker L, we apply row reduction to the matrix:  

1 0 − 1 1 2 − 1 1 0 − 1

  (^) →

 

1 0 − 1 0 2 0 0 0 0

  (^) →

 

1 0 − 1 0 1 0 0 0 0

 

Hence (x, y , z) ∈ ker L if x − z = y = 0. It follows that ker L is the line spanned by (1, 0 , 1).

More examples

f : M 2 , 2 (R) → M 2 , 2 (R), f (A) = A + AT^.

f

a b c d

2 a b + c b + c 2 d

ker f is the subspace of anti-symmetric matrices, the range of f is the subspace of symmetric matrices.

g : M 2 , 2 (R) → M 2 , 2 (R), g (A) =

  0 1 0 0

 A.

g

a b c d

c d 0 0

The range of g is the subspace of matrices with the zero second row, ker g is the same as the range =⇒ g (g (A)) = O.