Linear Space / Vector Space • A set of vectors • A set of numbers • Addition of vectors • Multiply vectors with numbers Zero Vector • There is a vector O such that for all vector x ○ x+O=x • Theorem ○ If O_1 and O_2 are both zero vectors, then O_1=O_2 • Proof ○ {█(O_1+O_2=O_1@O_2+O_1=O_2 )┤⇒O_1=O_2 Existence of Negative Vector • For every vector x, there is a vector y such that • x+y=0 • denoted as −x Multiplication with Numbers (Scalers) • x,y:vectors, s,t:numbers (Number field:Q,R,ℂ) • s(x+y)=sx+sy • (s+t)x=sx+tx • s(tx)=(st)x • 0⋅x=0 • 1⋅x=x Example of a Common Vector Spaces • R3={(x_1,x_2,x_3 )│x_1∈Rx_2∈Rx_3∈R is a vector space • Addition and multiplication defined as ○ (x_1,x_2,x_3 )+(y_1,y_2,y_3 )≝(x_1+y_1,x_2+y_2,x_3+y_3 ) ○ t(x_1,x_2,x_3 )≝(tx_1,tx_2,tx_3 ) Example of a Strange Vector Spaces • Number:R • Vector:R+=(0,∞) • Addition ○ x⨁y=x×y ○ e.g. √2⨁√2=√2×√2=2 ○ Zero vector: 1 • Inverse of Addition ○ Given x, find y ○ x⨁y=1 ○ ⇒y=1/x • Multiplication with numbers ○ t∈R, x∈R_+ ○ t⨀x≝x^t • Proof: Distributive law ○ t⨀(s⨀x)=(x^s )t=xst=(ts)⨀x
