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Definition of orthogonal vectors, sets, and subspaces; properties and benefits.

Recall that a set S is a basis for a subspace X if [ S ] = X and S has linearly independent elements. If S is a basis for X then each x X is uniquely determined by ( a 1 , a 2 , ... , a n ) such that i = 1 n a i s i = x . In this sense, we could operate either with x itself or with the vector a = [ a 1 , ... a n ] = R n . One would wonder then whether particular operations can be performed with a representation a instead of the original vector x .

Example 1 Assume x , y X have representations a , b R n in a basis for X . Can we say that x , y = a , b ?

For the particular example of X = L 2 [ 0 , 1 ] , S = { 1 , t , t 2 } so that [ S ] = Q , the set of all quadratic functions supported on [ 0 , 1 ] . Pick x = 2 + t + t 2 and y = 1 + 2 t + 3 t 2 . One can see then that if we label s 1 = 1 , s 2 = t , s 3 = t 2 , then the coefficient vectors for x and y are a = [ 2 1 1 ] and b = [ 1 2 3 ] , respectively. Let us compute both inner products:

x , y = 0 1 x ( t ) y ( t ) d t = 0 1 ( 2 + t + t 2 ) ( 1 + 2 t + 3 t 2 ) d t = 187 20 9 . 35 , a , b = 2 + 2 + 3 = 7 .

Since 7 9 . 35 , we find that we fail to obtain the desired equivalence between vectors and their representations.

While this example was unsuccessful, simple conditions on the basis S will yield this desired equivalence, plus many more useful properties.

Several definitions of orthogonality will be useful to us during the course.

Definition 1 A pair of vectors x and y in an inner product space are orthogonal (denoted x y ) if the inner product x , y = 0 .

Note that 0 is immediately orthogonal to all vectors.

Definition 2 Let X be an inner product space. A set of vectors S X is orthogonal if x , y = 0 for all x , y S , x y .

Definition 3 Let X be an inner product space. A set of vectors S X is orthonormal if S is an orthogonal set and s = s , s = 1 for all s S .

Definition 4 A vector x in an inner product space X is orthogonal to a set S X (denoted x S ) if x y for all y S .

Definition 5 Let X be an inner product space. Two sets S 1 X and S 2 X are orthogonal (denoted S 1 S 2 ) if x y for all x S 1 and y s 2 .

Definition 6 The orthogonal complement S of a set S is the set of all vectors that are orthogonal to S .

Benefits of orthogonality

Why is orthonormality good? For many reasons. One of them is the equivalence of inner products that we desired in a previous example. Another is that having an orthonormal basis allows us to easily find the coefficients a 1 , ... a n of x in a basis S .

Example 2 Let x X and S be a basis for X (i.e., [ S ] = X ). We wish to find a 1 , ... a n such that x = i = 1 n a i s i . Consider the inner products

x , s i = j = 1 n a j s j , s i = j = 1 n a j s j , s i ,

due to the linearity of the inner product in the first term. If S is orthonormal, then we have that for i j s j , s i = 0 . In that case the sum above becomes

x , s i = a i s i , s i = a i s i 2 = a i ,

due to the orthonormality of S . In other words, for an orthonormal basis S one can find the basis coefficients as a i = x , s i .

If S is not orthonormal, then we can rewrite the sum above as the product of a row vector and a column vector as follows:

x , s i = s 1 , s i s 2 , s i s n , s i a 1 a 2 a n .

We can then stack these equations for i = 1 , ... , n to obtain the following matrix-vector multiplication:

x , s 1 x , s 2 x , s n β = s 1 , s 1 s 2 , s 1 s n , s 1 s 1 , s 2 s 2 , s 3 s n , s 2 s 1 , s n s 2 , s n s n , s n G a 1 a 2 a n a .

The nomenclature given above provides us with the matrix equation β = G · a , where β and G have entries β i = x , s i and G i , j = s j , s i , respectively.

Definition 7 The matrix G above is called the Gram matrix (or Gramian) of the set S .

In the particular case of orthonormal S , it is easy to see that G = I , the identity matrix, and so a = β as given earlier. For invertible Gramians G , one could compute the coefficients in vector form as a = G - 1 β . For square matrices (like G ), invertibility is linked to singularity.

Definition 8 A singular matrix is a non-invertible square matrix. A non-singular matrix is an invertible square matrix.

Theorem 1 A matrix is singular if G · x = 0 for some x 0 . A matrix is non-singular if G · x = 0 only for x = 0 .

The link between this notion of singularity and invertibility is straightforward: if G is singular, then there is some a 0 for which G · a = 0 . Consider the mapping y = G · x ; we would also have y = G ( x + a ) . Since x x + a , one cannot “invert” the mapping provided by G into y .

Theorem 2 S is linearly independent if and only if G is non-singular (i.e. G x = 0 if and only if x = 0 ).

Proof: We will prove an equivalent statement: S is linearly dependent if and only if G is singular, i.e., if and only if there exists a vector x 0 such that G x = 0 .

( ) We first prove that if S is linearly dependent then G is singular. In this case there exist a set { a i } R , with at least one nonzero, such that i = 1 n a i s i = 0 . We then can write i = 1 n a i s i , s j = 0 , s j = 0 for each s j . Linearity allows us to take the sum and the scalar outside the inner product:

i = 1 n a i s i , s j = 0 .

We can rewrite this equation in terms of the entries of the Gram matrix as i = 1 n a i G j i = 0 . This sum, in turn, can be written as the vector inner product

G j 1 G 1 n a 1 a n = 0 ,

which is true for every value of j . We can therefore collect these equations into a matrix-vector product:

G 11 G 1 n G n 1 G n n a 1 a n = 0 0 .

Therefore we have found a nonzero vector a for which G a = 0 , and therefore G is singular. Since all statements here are equalities, we can backtrack to prove the opposite direction of the theorem ( ) .

Pythagorean theorem

There are still more nice proper ties for orthogonal sets of vectors. The next one has well-known geometric applications.

Theorem 3 (Pythagorean theorem) If x and y are orthogonal ( x y ), then x 2 + y 2 = x + y 2 .

Proof:

x + y 2 = x + y , x + y = x , x + x , y + y , x + y , y

Because x and y are orthogonal, x , y = y , x = 0 and we are left with x , x = x 2 and y , y = y 2 . Thus: x + y 2 = x 2 + y 2 .

Questions & Answers

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Maira Reply
what are the products of Nano chemistry?
Maira Reply
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learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Google
da
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Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
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RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
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Professor
I think
Professor
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Brian Reply
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Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
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What is meant by 'nano scale'?
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LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
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Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
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Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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Bob Reply
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Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
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Damian Reply
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Renato
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Kyle
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Adin
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biomolecules are e building blocks of every organics and inorganic materials.
Joe
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Source:  OpenStax, Introduction to compressive sensing. OpenStax CNX. Mar 12, 2015 Download for free at http://legacy.cnx.org/content/col11355/1.4
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