For the transformation L:C^1[a,b]->C[a,b] definied by L(f)=dfdx (means, derivative of function f w.r.t x)

Jaxson Mack 2022-08-07 Answered
For the transformation L : C 1 [ a , b ] C [ a , b ] definied by L ( f ) = d f d x (means, derivative of function f w.r.t x),
a) Show that L is linear.
b) Find K e r ( L ).
c) Find n u l l i t y ( L ).
d) Is L one-to-one? Explain.
e) is L invertible? Explain.
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Answers (2)

Siena Bennett
Answered 2022-08-08 Author has 17 answers
a) Consider, c be scalar and f , g c 1 [ a , b ]
L ( c f + g ) = d d x ( c f + g )
= d d x ( c f ) + d d x ( g )
= c d d x ( f ) + d d x ( g )
L ( c f + g ) = c L ( f ) + L ( g )
L is linear
b) K e r ( L ) = { f c 1 [ a , b ] : d f d x = 0 }
K e r ( L ) = { constant terms }
Basis for K e r ( L ) = { 1 }
c) N u l l i t y ( L ) = 1
d) As we know L is 1 1 iff K e r ( L ) = { 0 }
\therefore L is not 1 1 ( K e r ( L ) { 0 } )
c) Any transformation is invertible if it is both 1 1 and onto. As L is not 1 1
L is not invertible
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brasocas6
Answered 2022-08-09 Author has 3 answers

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