The notes cover linear operators and discuss linear independence of functions (Boas ).

Transcription

1 Linear Operators Hsiu-Hau Lin Mar 25, 2010 The notes cover linear operators and discuss linear independence of functions Boas Linear operators An operator maps one thing into another For instance, the ordinar functions are operators mapping numbers to numbers A linear operator satisfies the properties, OA + B OA + OB, OkA koa, 1 where k is a number As we learned before, a matri maps one vector into another One also notices that Mr 1 + r 2 Mr 1 + Mr 2, Mkr kmr Thus, matrices are linear operators Orthogonal matri The length of a vector remains invariant under rotations, M T M The constraint can be elegantl written down as a matri equation, M T M MM T 1 2 In other words, M T M 1 For matrices satisf the above constraint, the are called orthogonal matrices Note that, for orthogonal matrices, computing inverse is as simple as taking transpose an etremel helpful propert for calculations From the product theorem for the determinant, we immediatel come to the conclusion det M ±1 In two dimensions, an 2 2 orthogonal matri with determinant 1 corresponds to a rotation, while an 2 2 orthogonal

2 HedgeHog s notes March 24, matri with determinant 1 corresponds to a reflection about a line Let s come back to our good old friend the rotation matri, cos θ sin θ Rθ, R T 3 sin θ cos θ sin θ cos θ It is straightforward to check that R T R RR T 1 You ma wonder wh we call the matri orthogonal? What does it mean that a matri is orthogonal? to what?! Here comes the charming reason for the name Writing down the product R T R eplicitl, cos θ sin θ v1 v 1 v 1 v 2 1 0, 4 sin θ cos θ v 2 v 1 v 2 v we realize that an orthogonal matri contains a complete bases of orthogonal vectors in the same dimensions! Rotations and reflections in 2D Consider the rotation matri and the reflection about the -ais also called parit operator in the -direction, cos θ sin θ Rθ, P sin θ cos θ We can construct two operators b combining Rθ and P in different orders, C RθP, D P Rθ 6 One can check that det C det D 1 and the do not correspond to the usual rotations Carring out the matri multiplication, the operator C in eplicit matri form is C 7 sin θ cos θ To figure what the operator do, we can act C on unit vectors along - and -directions, 1 cos θ, sin θ cos θ 0 sin θ 0 sin θ sin θ cos θ 1 cos θ

3 HedgeHog s notes March 24, Plotting out the mappings, one can see that C corresponds to a reflection about the line at θ/2 While the geometric picture is nice, it is also comforting to know about the algebraic approach, Cr r sin θ cos θ 8 After some algebra, the above matri equation gives the relation for the reflection line, sinθ/2 cosθ/2 This is eactl what we epected Now we turn to the other operator D, 1 0 cos θ sin θ cos θ sin θ D P Rθ 0 1 sin θ cos θ sin θ cos θ You ma have guessed that D corresponds to a reflection about some line this is indeed true Absorbing the minus sign into the sin function, we come to the identit P Rθ R θp R 1 θp 9 Thus, D corresponds to a reflection about the line at θ/2 Rotations and reflections in 3D We can generalize the discussions to three dimensions An 3 3 orthogonal matrices with determinant 1 can be brought into the standard form b choosing the rational ais to coincide with the z-ais, Rθ cos θ sin θ 0 sin θ cos θ 0 10 Similarl, An 3 3 orthogonal matrices with determinant 1 can be brought into the standard form, Rθ cos θ sin θ 0 sin θ cos θ

4 HedgeHog s notes March 24, and corresponds to a rotation about the appropriate z-ais followed b a reflection through the -plane An eample will help to digest the notaion, L First of all, det L 1 and thus corresponds to an improper rotation rotation + reflection We can find out the normal vector for the reflection plane, Ln n n n n z Or, we can take a different view and tr to figure out the equation for the plane directl, Lr r z Both methods give the reflection plane + 0 and eplains the action of the operator L z n n n z Wronskian for linear independence Following similar definition for vectors, we sa that a set of functions is linearl dependent if some linear combinations of them give identical zero, k 1 f 1 + k 2 f k n f n 0, 12 where k k k 2 n 0 Taking derivatives of the above equation, we can cook up a complete set of equations, k 1 f 1 + k 2 f 2 + +k n f n 0, k 1 f 1 + k 2 f 2+ +k n f n 0, k 1 f n k 2 f n k n f n n 1 0

5 HedgeHog s notes March 24, If we can find non-trivial solutions for k 1, k 2,, k n, the functions are linearl dependent From previous lectures, we know that it amounts to require W f 1, f 2,, f n f 1 f 2 f n f 1 f 2 f n 0, 13 f n 1 1 f n 1 2 f n n 1 where W f 1, f 2,, f n is the Wronskian It is important to emphasize that dependent functions implies W 0, but W 0 does not necessaril impl the functions are linearl dependent