Name:-
Muhammad Hanzla
Student ID:-
S2019266005
Section:-
v1
Topic:-
Chapter 6
Professor:-
Shahid Imran
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Inner Space product, Assignments of Linear Algebra
Orthogonal vectors Inner space product
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Name:-
Muhammad Hanzla
Student ID:-
S
Section:-
v
Topic:-
Chapter 6
Professor:-
Shahid Imran
Presentation
Linear Algebra
Doctor Shahid Imran
Topic
Inner Dot Product
D(u, v)=||u-v||=sqrt(u-v,u-v)
Unit Vector:-
A vector of norm 1 is called a unit vector.
Weights:-
The Euclidean inner product is the most important inner product on R^n , there are various applications in which it is desirable to modify it by weighting each term differently. More precisely, if w1,w2,…….wn are positive real numbers, which we will call weights.
Weighted Euclidean inner product:-
If u=( u1,u2,……un ) and v=( v1,v2,……vn ) are vectors in R^n, then it can be shown that the formula (u, v)= ( w1u1v1+w2u2v2+……+wnunvn ) defines an inner product on R^n that we call the weighted Euclidean inner product with weights (^) w1,w2,…….wn. Unit Circles and Spheres in Inner Product Spaces:
Unit Circle:-
If V is an inner product space, then the set of points in V that satisfy ||u||= is called the unit sphere or sometimes the unit circle in V. Inner Products Generated by Matrices:
matrix inner products:-
The Euclidean inner product and the weighted Euclidean inner products are special cases of a general class of inner products on R^n called matrix inner products. To define this class of inner products, let u and v be vectors in R^n that are expressed in column form, and let A be an nvertible n*n matrix.
(u, v)=Au.Av And the formula we use in this section is {u,v}=(Av)^T.Au Concept Review:- Inner product axioms Euclidean inner product Euclidean n-space Weighted Euclidean inner product Unit circle (sphere) Matrix inner product Norm in an inner product space Distance between two vectors in an inner product space Examples of inner products Properties of inner products Skills:- Compute the inner product of two vectors. Find the norm of a vector. Find the distance between two vectors. Show that a given formula defines an inner product. Show that a given formula does not define an inner product by demonstrating that at least one of the inner product space axioms fails.
Orthogonality-:
a.b=(1)(2)+(2)(-1)+(0)(10) a.b=2-2+ a.b= So these two vectors is also orthogonal vectors
- Prove that vectors a=(2,3,1) and b=(3,1,-9) is orthogonal vectors:- a.b=(2)(3)+(3)(1)+(1)(-9) a.b=6+3- a.b= So these two vectors is also orthogonal vectors.
- Are these vectors a=(3,-1) and b=(7,5) is orthogonal or not? a.b=(3)(7)-(1)(5) a.b=21- a.b= Since their dot product is not equal to zero so these two vectors are not orthogonal. Orthogonal Sets:- A nonempty set in vectors R^n is called an orthogonal sets if all pair of the distinct vectors in the set is orthogonal. For example:- i.j=i.k=j.k=0 is the example of orthogonal sets. Orthonormal Set:- An orthogonal set of a unit vector is called orthonormal set. Orthogonal Projection:- It is necessary to decompose a vector into a sum of two vectors one term being a scalar multiple of a specified a nonzero vector and other being called a orthogonal. You will clearly understand orthogonal projection from this image
Concept Review:- Cauchy–Schwarz inequality Angle between vectors Orthogonal vectors Orthogonal complement Orthogonal and orthonormal sets Normalizing a vector Orthogonal projections Skills:- Find the angle between two vectors in an inner product space. Determine whether two vectors in an inner product space are orthogonal. Find a basis for the orthogonal complement of a subspace of an inner product space. Determine whether a set of vectors is orthogonal (or orthonormal).