Let T :V -> W be a one to one linear transformation. Show that the image of a linearly independent set of vectors in V is a linearly independent set in W
Solution
let v 1 , ..., v n be a linearly independent set of vectors in V.
since these are LI, by definition:
if a 1 v 1 + a 2 v 2 + ... + a n v n = 0, then every a i = 0.
then if, : a 1 w 1 + a 2 w 2 + ... + a n w n = 0, we have to show that all a i = 0,
a 1 w 1 + a 2 w 2 + ... + a n w n = a 1 T(v 1 ) + ... + a n T(v n ) = T(a 1 v 1 + a 2 v 2 + ... + a n v n ) = 0
since T is 1-1, T(v) = 0 implies v = 0,
so: a 1 v 1 + a 2 v 2 + ... + a n v n = 0
which implies every a i = 0,
there fore the w\'s are linearly independent, by definition.
.
Python Notes for mca i year students osmania university.docx
Let T -V -- W be a one to one linear transformation- Show that the ima.docx
1. Let T :V -> W be a one to one linear transformation. Show that the image of a linearly
independent set of vectors in V is a linearly independent set in W
Solution
let v 1 , ..., v n be a linearly independent set of vectors in V.
since these are LI, by definition:
if a 1 v 1 + a 2 v 2 + ... + a n v n = 0, then every a i = 0.
then if, : a 1 w 1 + a 2 w 2 + ... + a n w n = 0, we have to show that all a i = 0,
a 1 w 1 + a 2 w 2 + ... + a n w n = a 1 T(v 1 ) + ... + a n T(v n ) = T(a 1 v 1 + a 2 v 2 + ... + a n v n ) = 0
since T is 1-1, T(v) = 0 implies v = 0,
so: a 1 v 1 + a 2 v 2 + ... + a n v n = 0
which implies every a i = 0,
there fore the w's are linearly independent, by definition.
