# 207 - w = f (u) w = f (v) Example: with Example: with where...

This preview shows pages 1–4. Sign up to view the full content.

1 w = f ( u ) w = f ( v ) Example: with Example: with where

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
2 Example: with T A ( u ) is the projection of u R 3 on to the xy -plane, which is the range of T A . Example: with shear transformation k = 2 Proof By the properties of the matrix-vector multiplication. Two special linear transformations: (1) the identity transformation I : R n R n with I ( x ) = x x R n . (2) the zero transformation T 0 : R n R m with T 0 ( x ) = 0 x R n . 7
3 There are functions from R n to R m satisfying condition (ii) of the definition of linear transformation but not (i). For example, 2 22 for 0 otherwise x xy x T y xy y ⎡⎤ ⎛⎞ ⎢⎥ = + ⎜⎟ ⎣⎦ ⎝⎠ 0 If a function T : R n R m satisfies condition (i), then for all integers k , l > 0 and x R n , T ( k x ) = T ( x + + x ) = T ( x ) + + T ( x ) = kT ( x ) and T ( x ) = T ((1/ l ) x + + (1/ l ) x ) = T ((1/ l ) x ) + + T ((1/ l ) x ) = lT ((1/ l ) x ), or (1/ l ) T ( x ) = T ((1/ l ) x ) . Also, T ( 0 ) = T ( 0 + 0 ) = T ( 0 ) + T ( 0 ), so T ( 0 ) = 0 . And T ( 0 ) = T ( x + ( 1) x ) = T ( x ) + T ( x ), so T ( x ) = T ( x ).

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 10/16/2010 for the course EE 155 taught by Professor Fong during the Spring '09 term at National Taiwan University.

### Page1 / 5

207 - w = f (u) w = f (v) Example: with Example: with where...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online