Vectors

Question 1

Axioms: Complex Vector Space

Answer 1

note (|vi means |v>)
I |vi + |wi = |wi + |vi
II (|vi + |wi) + |ui = |vi + (|wi + |ui)
III ∃ |0i with (∀|vi) (|vi + |0i = |vi)
IV ∃ |v¯i with (|v¯i + |vi = |0i)
V 1|vi = |vi
VI λ(µ|vi) = (λµ)|vi
VII λ(|vi + |wi) = λ|vi + λ|wi
VIII (λ + µ)|vi = λ|vi + µ|vi

Question 2

Definition: Linear Combination of Vectors

Answer 2

Sum with complex and vectors stuff

Question 3

Theorem: Cancellation Theorem

Answer 3

|vi + |v1i = |vi + |v2i
then
|v1i = |v2i

Question 4

Theorem: Cancellation Theorem Corollary

Answer 4

vi + |wi = |vi or |wi + |vi = |vi
then
|wi = |0i

Question 5

Theorem: Zero times any vector equals zero

Answer 5

0|vi = |0i

Question 6

Theorem: Zero vector with scalar multiplication

Answer 6

λ|0i = |0i

Question 7

if λ != 0 and λ|vi = |0i

Answer 7

|vi = |0i.

Question 8

if |vi != |0i, and λ|vi = |0i

Answer 8

λ = 0.

Question 9

Theorem: Unique additive inverse

Answer 9

Inverses are unique

Question 10

Theorem: Additive inverse property

Answer 10

Inverses are the original but negative ie -|vi

Question 11

Theorem: Equivalent coefficients

Answer 11

|vi != 0 and λ|vi = µ|vi =⇒ λ = µ