 
Summary: Com S 633: Randomness in Computation
Lecture 17 Scribe: Ankit Agrawal
1 Vector Spaces
Denition: A vector over < n is a ntuple V =< x 1 ; x 2 ; :::; x n > where each x i 2 <. Hence,
a vector can be understood to be a line from the origin to a ndimensional point in the
ndimensional space.
1.1 Operations on Vectors
Multiplication by a real number:
Let a 2 <, V 2 < n , V =< x 1 ; x 2 ; :::; x n >, then
aV =< ax 1 ; ax 2 ; :::; ax n >
Addition of two vectors:
Let V =< x 1 ; x 2 ; :::; x n >, U =< y 1 ; y 2 ; :::; y n >, then
V + U =< x 1 + y 1 ; x 2 + y 2 ; :::; x n + y n >
Norms of a vector:
Let V =< x 1 ; x 2 ; :::; x n >, then
kV k 1 = L 1 norm of V =
n
X
i=1
jx i j
