Finding Inner Product, Length, and Distance In Exercises 17-26, find (a) 〈 u , v 〉 , (b) ‖ u ‖ , (c) ‖ v ‖ , and (d) d ( u , v ) for the given inner product defined on R n . u = ( 8 , 0 , − 8 ) , v = ( 8 , 3 , 16 ) , 〈 u , v 〉 = 2 u 1 v 1 + 3 u 2 v 2 + u 3 v 3
Finding Inner Product, Length, and Distance In Exercises 17-26, find (a) 〈 u , v 〉 , (b) ‖ u ‖ , (c) ‖ v ‖ , and (d) d ( u , v ) for the given inner product defined on R n . u = ( 8 , 0 , − 8 ) , v = ( 8 , 3 , 16 ) , 〈 u , v 〉 = 2 u 1 v 1 + 3 u 2 v 2 + u 3 v 3
Solution Summary: The author explains that the value of inner product langle u,vrangle is 0.
Finding Inner Product, Length, andDistance In Exercises 17-26, find (a)
〈
u
,
v
〉
, (b)
‖
u
‖
, (c)
‖
v
‖
, and (d)
d
(
u
,
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for the given inner product defined on
R
n
.
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=
(
8
,
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,
−
8
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,
v
=
(
8
,
3
,
16
)
,
〈
u
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=
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3
(a) Let V be R², and the set of all ordered pairs (x, y) of real numbers.
Define an addition by (a, b) + (c,d) = (a + c, 1) for all (a, b) and (c,d) in V.
Define a scalar multiplication by k · (a, b) = (ka, b) for all k E R and (a, b) in V.
.
Verify the following axioms:
(i) k(u + v) = ku + kv
(ii) u + (-u) = 0
Exercise 4 (Proving a fact about Cartesian products). Let X CR² be a cartesian
product of two subsets of R, i.e. there exist C, DCR such that X = C x D.
Prove (rigorously) that if {(-3, 1), (0, -2), (6, 2)} CX, then X contains at least
nine points. Which ones?
Using basis and dimensions in vector space section for linear algebra
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