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))±.