Chapter 5 - Inner Product Spaces - 5.1 Definition of an Inner Product Space - Problems - Page 350: 14

See below

Given: $p_1(x)=a+bx\\ p_2(x)=c+dx$ be vectors in $P_1(R)$ 1. $

=aa+bb=a^2+b^2 \gt 0$ If $p(x)=0+0x \rightarrow

=0^2+0^2=0$ If $p(x) \ne 0 \rightarrow

=a^2+b^2 \gt 0$ if $a \ne 0, a^2 \gt 0 ;b \ne 0, b^2 \gt0$ Hence, $

=0$ if and only if $p(x)=0+0x$ 2. $

=ac+bd=ca+db=$ 3. With scalar $k$, we have $kp_1(x)=ka+kbx$ and $=(ka)c+(kb)d\\ =kac+kbd\\ =k(ac+bd)\\ =k$ 4. Let $p_3(x)=e+fx$ in $P_1(R)$ $p_1(x)+p_2(x)=(a+c)+(b+d)x\\ \rightarrow =(a+c)e+(b+d)f\\ =ae+ce+bf+df\\ =(ae+bf)+ce+df\\ =+$