4. Let Vheavectorspaee overaeld F. Ifisaxednonnero vector andcaisascalar,
then the linear transformation 9 : V 1- V dened by 9(52' = of is bijective.
5. If T : V -> W is a linear transformation and cfw_551, E2, in are linearly dependent
vectors in V, then
1. In each of the examples below, determine which eld axioms are valid and which are not.
Which examples are elds? In each case that an axiom fails to hold, give an example to
show why it tails to hold.
(a) (.3, +? J wh
Delaware State University
Linear Algebra Theory
Dr. Dawn A. Lotts Personal Notes
Please report any errors in this document to Dr. Lott at [email protected]
Table of Contents
1 Mathematical Logic 
1.1 Propositions . . . . . . . . . . .
For each of these, when the subset is independent it must be proved, and when the subset is dependent an example ol a dependence must be given.
L It is dependent. Corsidehng
1 2 4 D
C] 3 -l- 02 2 + cs -4 = D
5 4 14 0
gives rise to this linear systems
Theorem 4.2. Let KW be vector spaces over a: eld F. Let T: V > W be a .linear
tmnsfonnation which is one-to-one. Assume that cfw_911, . . . mg is linearly independent. Then
cfw_11111. . . . ,Tfvn is also linearly independent.
Proof. We argue by contra