{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

14_hwc1SolnsODDA

# 14_hwc1SolnsODDA - 2.23 Let f be a linear transformation...

This preview shows page 1. Sign up to view the full content.

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

Unformatted text preview: 2.23 Let f be a linear transformation from IR2 to IR2 such that f 1 1 = a 3 1 2 and f 2 1 = 1 2a Find all values of a, if any, for which f = 5 . 5 SOLUTION If we can express 1 2 as a linear combination of 1 1 2 =s +t 2 1 1 1 1 and 2 , say 1 Then we can use the linearity of f to compute f 1 2 . There are several ways to do this. The simplest in IR2 is to express one of the standard basis vectors as a linear combination of the given vectors. This is easy enough: 1 2 1 = − 0 1 1 Hence, the the linearity of f , f 1 0 =f = 2 1 −f 1 1 . 1 a 1−a − = 2a 3 2a − 3 Now it is easy: As you see, 1 2 1 1 1 =2 − , and so, by the linearity of f , 2 1 0 = 2f =2 1 1 −f 1 0 f a 1−a 3a − 1 − = 3 2a − 3 9 − 2a 3a − 1 5 = 9 − 2a 5 and 15 which is equivalent to the pair of Therefore, f equations 1 2 = 5 5 reduces to 3a − 1 = 5 9 − 2a = 5 . 21/september/2005; 22:09 ...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online