Principles of Mathematics in Operations Research

3.1 Inner Products 35

Remark 3.1.5 \\x\\l geometrically amounts to the Pythagoras formula ap-

plied (n-1) times.

Definition 3.1.6 The quantity xTy is called inner product of the vectors x

and y in K"

n

xTy = ^x^ji.

»=i

Proposition 3.1.7

xTy = 0#ilj.

Proof. (<=) Pythagoras Formula: ||x|| + ||y|| = ||a; — y\\ ,

\\x ~ y\\^2 = T,7=i(xi ~Vi)^2 = \\x\? + \\y\\^2 - 2xTy- The last tw0 identities yield

the conclusion, xTy = 0.

(=») xTy = 0 =*- II^H^2 + \\yf = \\x - y\\^2 =>x±y. •

Theorem 3.1.8 (Schwartz Inequality)

\xTy\ < \\x\\ 2 \\y\\ 2 , x,y£Rn.

Proof. The following holds Va € R:

0 < ||x + ay\\l =xTx + 2 \a\ xTy + o?yTy = \\x\\^22 + 2 \a\ xTy + a^2 \\y\\^22 , (*)

Case (x A. y): In this case, we have =>• xTy = 0 < \\x\\ 2 \\y\\ 2.

Case (x JL y): Let us fix a = l^f. Then, (*) 0 < - ||a;||^2 + 'ffljffi. •

3.1.2 Orthogonal Spaces

Definition 3.1.9 Two subspaces U and V of the same space R™ are called

orthogonal ifMu 6 J/,Vu G V, u Lv.

Proposition 3.1.10 Af(A) andlZ(AT) are orthogonal subspaces of W,M(AT)

and H(A) are orthogonal subspaces of Km.

Proof. Let w G M(A) and v G H(AT) such that Aw = 6, and v = ATx for
some x G R". wTv = wT(ATx) = (wTAT)x — 9 Tx — 0. •

Definition 3.1.11 Given a subspace V o/Rn, the space of all vectors orthog-
onal to V is called the orthogonal complement of V, denoted by V^1 -.

Theorem 3.1.12 (Fundamental Theorem of Linear Algebra, Part 2)

Af(A) = (n(AT))^, K(AT) = (Af(A))\

Af(AT) = (K(A))^, 11(A) = (Af(AT))±.