Guided Proof Prove that if w is orthogonal to each vector in S = { v 1 , v 2 , … , v n } , then w is orthogonal to every linear combination of vector in S . Getting Started: To prove that w is orthogonal to every linear combination of vectors in S , you need to show that their inner product is 0 . (i) Write v as a linear combination of vectors, with arbitrary scalars c 1 , … , c n in S . (ii) Form the inner product of w and v . (iii) Use the properties of inner products to rewrite the inner product 〈 w , v 〉 as a linear combination of the inner products 〈 w , v i 〉 , i = 1 , … , n . (iv) Use the fact that w is orthogonal to each vector in S to lead to the conclusion that w is orthogonal to v .
Guided Proof Prove that if w is orthogonal to each vector in S = { v 1 , v 2 , … , v n } , then w is orthogonal to every linear combination of vector in S . Getting Started: To prove that w is orthogonal to every linear combination of vectors in S , you need to show that their inner product is 0 . (i) Write v as a linear combination of vectors, with arbitrary scalars c 1 , … , c n in S . (ii) Form the inner product of w and v . (iii) Use the properties of inner products to rewrite the inner product 〈 w , v 〉 as a linear combination of the inner products 〈 w , v i 〉 , i = 1 , … , n . (iv) Use the fact that w is orthogonal to each vector in S to lead to the conclusion that w is orthogonal to v .
Solution Summary: The author explains that the vector v is a linear combination of vectors, with arbitrary scalars in S.
Guided Proof Prove that if
w
is orthogonal to each vector in
S
=
{
v
1
,
v
2
,
…
,
v
n
}
, then
w
is orthogonal to every linear combination of vector in
S
.
Getting Started: To prove that
w
is orthogonal to every linear combination of vectors in
S
, you need to show that their inner product is
0
.
(i)
Write
v
as a linear combination of vectors, with arbitrary scalars
c
1
,
…
,
c
n
in
S
.
(ii)
Form the inner product of
w
and
v
.
(iii)
Use the properties of inner products to rewrite the inner product
〈
w
,
v
〉
as a linear combination of the inner products
〈
w
,
v
i
〉
,
i
=
1
,
…
,
n
.
(iv)
Use the fact that
w
is orthogonal to each vector in
S
to lead to the conclusion that
w
is orthogonal to
v
.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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