8.5 Series of Products 499- Vx\ '[/,TE !Rn, r + ('[/ + 7) = (r + '[/) + 7.
- Vr, '[/ E !Rn, r + '[/ = '[/ + r.
- 3 (}' E !Rn 3 Vr E !Rn, r + (}' = (}' + r = r. [Q' = (0,0,0, · · · ,0)].
- Vr E !Rn, 3-r E !Rn 3 r+-r = -=-' 0. [-r = (-x1,-X2,-X3,··· ,-xn)].
- Vr E !Rn, '<Ir E IR, rr E !Rn.
- Vr,'[/EIRn, VrEIR, r(r+'[/)=rr+r'[/.
8.VrEIRn,vr,sEIR, (r+s)r=rr+sr.
- Vr E !Rn, Vr, s E IR, (rs)r = r(sr).
- Vr E !Rn, lr = r.
Proof. Consult any standard textbook in elementary linear algebra. •Because !Rn has these properties it is called a vector space. All of the
algebraic properties of a general vector space are derivable from these proper-
ties. They are well known from your linear algebra course and are not repeated
here.In !Rn there is a kind of product often called an "inner product." Specifi-
cally, we define the dot product of two n-vectors rand'[/ in !Rn as the sum
of the products of their components:
= X1Y1 + X2Y2 + X3y3+ · · · + XnYn
n
= I: XiYi·
i=lExample 8.5.12 (5, 2, -4, 1) · (3, -7, 0, -8) = 15 - 14 - 0 - 8 = -7.
The following theorem lists the basic algebraic properties of the dot product
in !Rn.
Theorem 8.5.13 Vr, '[/,TE !Rn, and Vr E IR,
i. r·'f/='f/·r
- (r + 'fl). 7 = (r. 7) + ('fl. 7)
- (rr). '[/ = r(r. '[/)
- r · r ~ O; moreover, r · r = 0 ¢:> r = 0.
Proof. Consult any standard textbook in elementary linear algebra. •All of the algebraic properties of the dot product are derivable from the
four properties listed in Theorem 8.5.13. Since their proofs are basic in any