= 9/20 연습 내용 = ”’고쳐놓았어요. – 김영욱”’
\[ 16. ~\text{ Suppose that u,v, and w are vectors such that} =2, = -3, = 5,|u| = 1, \] \[ |v|=2, |w|=7 \] ~$$ ~\text{ Evaluate the given expression. }~~ \text{(a)} ~< u+v,v+w> ~~~ \text{(b)}~~ < 2v-w ,3u+2w> $$
Ans of (a) : \[ = + + + = 2 + 5 +4 -3 = 8 . \]
Ans of (b) : \[ = 6 +4 -3 -2 = 12 -12 -15-98 = -113 . \]
\[ 21. ~\text{ Show that the following identity holds for vectors in any inner product space.} \]
\[ = {\frac14} |u+v|^2 – \frac14 |u-v|^2 \]
Ans : \[ Since~~ |u+v|^2 = = + + + \] \[ Since~~ |u-v|^2 = = – – + \] \[ Thus~~~ |u+v|^2 – |u-v|^2 = 4 . \]
\[ 24. ~\text{ Prove : If } \text{is the Euclidean inner product on}~ R^n ,\text{ and if}~ A~ \text{is an} \] n×n \[ \text{matrix , then} \] \[ = \]
Ans : \[ \text{Since} = (Av)^T u = v^T A^T u = v^T (A^T u) = . \]
\[ 27. ~\text{ Use the inner pruduct to compute} \] \[
\text{for the vectors p=p(x) and q=q(x) in} P_3 \]
\[ ~~~~~~~~~~~~~~~
= \int_{-1}^1~~ p(x)q(x)~ dx \]
Ans : Check first this is inner product.
① \[
= \int_{-1}^1~~ p(x)q(x)~ dx ~~~~~~ = \int_{-1}^1~~q(x)p(x) ~ dx \]
② \[
= \int_{-1}^1~~ p(x)[q(x)+r(x)]~ dx = \int_{-1}^1~~p(x)q(x)~dx ~~+\int_{-1}^1~~p(x)r(x)~dx ~~then ~
=
+
~ \]
③ \[ = \int_{-1}^1~~ [kp(x)]q(x)~ dx ~~=k\int_{-1}^1~~ p(x)q(x)~ dx ~~\]
④ \[
= \int_{-1}^1~~ p(x)^2~ dx~ \geq 0 ~~ \text{Since} ~~ p(x)^2 \geq 0 ~~~ \text{and}~~
=0 ~~\text{iff}~~ p(x)=0 \]
And we sloved the problem ”’31”’.
\[ 19. \text{ Let V be an inner product space. Show that if}~ u~\text{ and}~ v \text{ are orthogonal unit vectors in}~ V, \text{ then}~ |u-v| = \]
Ans : \[ \text{ Since }~ u \text{ and}~ v \text{ are orthogonal, then} \] \[ = o ~\text{and Since}~ u \text{ and}~ v \text{ are a unit then}~ |u|=|v|=1 \] \[ \text{Thus}~ |u-v|^2 = = – 2 + = 1+1 =2 . \]
\[ 24. \text{ Prove the following generalization of Theorem 6.2.4. If }~v_1, v_2 , \dots , v_r \text{ are pairwise orthogonal vectors in an inner product space in V,} \] \[ \text{ then} ~ |v_1 + v_2 + \dots + v_r |^2 = |v_1|^2 + \dots +|v_r|^2 \]
Ans : \[ \text{ Since}~ v_i’s \text{ are pairwise orthogonal, then}~ =0 \text{ if } i\neq j . \text{thus proved.} \]