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.)

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Chapter2: Second-order Linear Odes
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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.)
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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