Linear Algebra - VectorsPlease help me with this question! Thank you! A set of vectors {u1,... , um} in R" is called orthonormal if1 if i = j0if i + j.U; · U;=In other words, an orthornormal set of vectors consists of unit vectors, such that eachvector 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 thatv = c¡Uj ++ CnUn•(1)..Show that any vector v E R" can be written asv = proju, (v) ++ proj, (v).(Hint: Try taking the dot product of certain vectors with both sides of Equation (1)above.)

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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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Linear Algebra - Vectors

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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.)

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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