Let {u1,..., un} be an orthonormal set in R". It is a fact that for any vector v E R", there exist scalars c1, ·.., Cn E R such that v = c¡U1+…+ CnUn. (1) Show that any vector v E R" can be written as v = proju, (v) + + proju, (v). ... (Hint: Try taking the dot product of certain vectors with both sides of Equation (1) above.)
Let {u1,..., un} be an orthonormal set in R". It is a fact that for any vector v E R", there exist scalars c1, ·.., Cn E R such that v = c¡U1+…+ CnUn. (1) Show that any vector v E R" can be written as v = proju, (v) + + proju, (v). ... (Hint: Try taking the dot product of certain vectors with both sides of Equation (1) above.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please help me with part b! Thank you!
![A set of vectors {u1,... , um} in R" is called orthonormal if
1 if i = j
0if i + j.
U; · U;
=
In other words, an orthornormal set of vectors consists of unit vectors, such that each
vector is orthogonal to all other vectors in the set.
(a) (i) For ui :=
in R2.
[}, ] and u2 :=
1, show that {u1, u2} is an orthonormal set
(ii) Fix [x, y] E R². Find scalars a, b e R such that
[æ, y] = a proju, ([a, y]) + b proju, ([x, y]).
(b) Let {u1,..., un} be an orthonormal set in R". It is a fact that for any vector v E R",
there exist scalars c1, ... , Cn E R such that
v = c¡Uj +
+ CnUn•
(1)
..
Show that any vector v E R" can be written as
v = proju, (v) +
+ proj, (v).
(Hint: Try taking the dot product of certain vectors with both sides of Equation (1)
above.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9ee89080-ea45-47d2-be13-9cc64e04f9b1%2Fe5f0adad-82f1-4f48-813e-6d2837885de4%2Fecunh5t_processed.png&w=3840&q=75)
Transcribed Image Text:A set of vectors {u1,... , um} in R" is called orthonormal if
1 if i = j
0if i + j.
U; · U;
=
In other words, an orthornormal set of vectors consists of unit vectors, such that each
vector is orthogonal to all other vectors in the set.
(a) (i) For ui :=
in R2.
[}, ] and u2 :=
1, show that {u1, u2} is an orthonormal set
(ii) Fix [x, y] E R². Find scalars a, b e R such that
[æ, y] = a proju, ([a, y]) + b proju, ([x, y]).
(b) Let {u1,..., un} be an orthonormal set in R". It is a fact that for any vector v E R",
there exist scalars c1, ... , Cn E R such that
v = c¡Uj +
+ CnUn•
(1)
..
Show that any vector v E R" can be written as
v = proju, (v) +
+ proj, (v).
(Hint: Try taking the dot product of certain vectors with both sides of Equation (1)
above.)
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