30-functionspace - FUNCTION SPACES AND LINEAR MAPS Homework 4.2 28,40,34,58,66,78 Math 21b O Knill EXAMPLE Find the eigenvectors to the eigenvalue of

30-functionspace - FUNCTION SPACES AND LINEAR MAPS Homework...

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FUNCTION SPACES AND LINEAR MAPS Math 21b, O. Knill Homework: 4.2: 28,40,34,58,66,78* FUNCTION SPACES. Functions on the real line can be added f + g scaled λf and contain a ”zero vector” 0. P n , the space of all polynomials of degree n . The space P of all polynomials. C ( R ), the space of all smooth functions on the line C ( T ) the space of all 2 π periodic functions. In all these function spaces, the function f ( x ) = 0 which is constantly 0 is the zero function. LINEAR TRANSFORMATIONS. A map T on a linear space X is called linear if T ( x + y ) = T ( x ) + T ( y ) , T ( λx ) = λT ( x ) and T (0) = 0. Examples are Df ( x ) = f 0 ( x ) on C Tf ( x ) = R x 0 f ( x ) dx on C . Tf ( x ) = f (2 x ). Tf ( x ) = sin( x ) f ( x ) on C Tf ( x ) = 5 f ( x ) Tf ( x ) = f ( x - 1). SUBSPACES, EIGENVALUES, BASIS, KERNEL, IMAGE are defined as before X linear subspace f, g X, f + g X, λf X, 0 X . T linear transformation T ( f + g ) = T ( f ) + T ( g ) , T ( λf ) = λT ( f ) , T (0) = 0. f 1 , f 2 , ..., f n linear independent i c i f i = 0 implies f i = 0. f 1 , f 2 , ..., f n span X Every f is of the form i c i f i . f 1 , f 2 , ..., f n basis of X linear independent and span. T has eigenvalue λ Tf = λf kernel of T { Tf = 0 } image of T { Tf | f X } . Some concepts do not work without modification. Example: det( T ) or tr( T ) are not always defined for linear transformations in infinite dimensions. The concept of a basis in infinite dimensions also needs to be defined

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