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Linear Operators

Module by: Richard Baraniuk

Summary: Introduction to linear operators.

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We will focus on linear operators in 301.

Definition 1: linear
An operator OO is linear if
Figure 1
Figure 1 (zero.png)
  • Superposition:: Ox1+x2=Ox1+Ox2 O x 1 x 2 O x 1 O x 2 . Figure 2 then Figure 3.
Figure 2
(a) (b)
Figure 2(a) (x1.png)Figure 2(b) (x2.png)
Figure 3
Figure 3 (sum.png)
  • Scaling:: Oγx=γOx O γ x γ O x , where γ∈ℂ γ . Figure 4 then Figure 5.
Figure 4
Figure 4 (x.png)
Figure 5
Figure 5 (gamma.png)

We can combine requirements 2 and 3 into:

O γ 1 x1+ γ 2 x2= γ 1 Ox1+ γ 2 Ox2 O γ 1 x 1 γ 2 x 2 γ 1 O x 1 γ 2 O x 2 (1)
Figure 6 is equivalent to Figure 7.
Figure 6
Right Hand Side
Right Hand Side (RHS.png)
Figure 7
Left Hand Side
Left Hand Side (LHS.png)

Exercise 1

Check for linearity/nonlinearity.

1.b)

Squarer

Fact 1

All matrix multiply operators are linear.

y=Ax y A x (2)
where x∈ℂN x N , y∈ℂN y N , and A A is N×N N N (Figure 8).
Figure 8
Figure 8 (systems.png)

Proof

Verify the 3 conditions for

y=∑k=0N−1akxk y k 0 N 1 a k x k (3)
where ak a k are the columns of matrix AA, A=a0a1…a N - 1 A a 0 a 1 … a N - 1 .
  1. Let x=0 x 0 , then all xk=0 x k 0 . This implies that y=0 y 0 .
  2. Show Ax1+x2=Ax1+Ax2 A x 1 x 2 A x 1 A x 2 .
    LHS=∑k=0N−1ak x 1 k+ x 2 k=∑k=0N−1ak x 1 k+∑k=0N−1ak x 2 k=Ax1+Ax2=RHS LHS k 0 N 1 a k x 1 k x 2 k k 0 N 1 a k x 1 k k 0 N 1 a k x 2 k A x 1 A x 2 RHS (4)
  3. Show Aγx=γAx A γ x γ A x
    LHS=∑k=0N−1akγxk=γ∑k=0N−1akxk=γAx=RHS LHS k 0 N 1 a k γ x k γ k 0 N 1 a k x k γ A x RHS (5)

Fact 2

Only matrix multiply operators are linear.

Upshot of Facts 1 and 2: The Study of linear operators in ℂN ℂ N is equivalent to the study N×N N N matrices.

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