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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A set of vectors {u1,... , Um} in R" is called orthonorma 1 if i = j 0 if i + j. 1 %3D U¡ · Uj

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter1: Vectors

Section1.3: Lines And Planes

Problem 47EQ

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Concept explainers

Magnitude

A vector is an element with both direction and magnitude. A vector is represented using an arrow, where the arrow represents the direction of the vector and its length is represented using the magnitude. The set of vectors in space is known as vector space. The vectors can be added, subtracted, multiplied, and divided in a vector space. The image below shows a vector AB.

Question

Linear Algebra - Vectors

Please help me with this question! 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.)

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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