Orthogonal projection, orthonormal basis, coordinate vector of the polynomial?

In summary, the conversation discussed finding the coordinate vector of a polynomial with respect to a given basis, finding an orthonormal basis for a given linear span, and finding the orthogonal projection of a vector on a subspace using the Gram-Schmidt algorithm.
  • #1
belleamie
24
0
Hey there I'm working on questions for a sample review for finals I'm stuck on these three I think I'm starting to confuse all the different theorem, I'm so lost please help

1) Find the coordinate vector of the polynomial
p(x)=1+x+x^2

relative to the following basis of P2:
p1=1+x, p2=1-x, p3=1+2x+3x^2

?
I wasnt sure how to work this problem out:
Does it start out as?
b1=1,t,t^2
b2=t,1,t^2
b3= 1+t, 1-t, t-t^2

2) Let X be the linear span of the vectors
(1,1,1,1) (1,1,1,0) (1,1,0,0)
in R^4. Find the orthonormal basis for X?

It is:
[[u1]]^2
[[u2]]^2
[[u3]]^2

u1=1/2(1,1,1,1)
u2=1/6(1,1,1,0)
u3=1/4(1,1,0,0)

3) Let X be the linear span of the vectors
(1,2,1,2) (1,2,1,0) (1,1,0,0)
in R^4. Find the orthogonal projection of the vector (1,1,1,1) on th esubspace X?
It is solved like this:
c1=(v,u1)/(u1/u2)=(1+2+1+2)/(1+4+1+4)
c2=(v,u2)/(u2/u2)=(1+2+1+0)/(1+4+1+0)
c3=(v,u3)/(u3/u3)=(1+1+0+0)/(1+1+0+0)

there for x=proj(v,x) = c1u1+c2u2+c3u3
 
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  • #2
belleamie said:
1) Find the coordinate vector of the polynomial
p(x)=1+x+x^2

relative to the following basis of P2:
p1=1+x, p2=1-x, p3=1+2x+3x^2

You want to find the vector [a,b,c] where p(x)=a*p1(x)+b*p2(x)+c*p3(x). This is the coordinate vector of p(x) with respect to your basis. I don't understand what followed, with the b's.

belleamie said:
2) Let X be the linear span of the vectors
(1,1,1,1) (1,1,1,0) (1,1,0,0)
in R^4. Find the orthonormal basis for X?

It is:
[[u1]]^2
[[u2]]^2
[[u3]]^2

u1=1/2(1,1,1,1)
u2=1/6(1,1,1,0)
u3=1/4(1,1,0,0)

An orthonormal basis is made up of orthogonal unit vectors.
Do you know the Gram-Schmidt orthogonalization algorithm? Use it to find an orthogonal basis, then make them unit vectors by dividing by their norms. Alternatively, you might be able to find an orthogonal basis by staring long enough, but Gram-Schmidt will work.

belleamie said:
3) Let X be the linear span of the vectors
(1,2,1,2) (1,2,1,0) (1,1,0,0)
in R^4. Find the orthogonal projection of the vector (1,1,1,1) on th esubspace X?
It is solved like this:
c1=(v,u1)/(u1/u2)=(1+2+1+2)/(1+4+1+4)
c2=(v,u2)/(u2/u2)=(1+2+1+0)/(1+4+1+0)
c3=(v,u3)/(u3/u3)=(1+1+0+0)/(1+1+0+0)

there for x=proj(v,x) = c1u1+c2u2+c3u3

In order to use that formula for the projection, you must use an orthogonal basis. Use Gram-Schmidt to get one.
 
  • #3

= (1/2,1/2,1/2,1/2)

1) The coordinate vector of a polynomial relative to a basis is a representation of the polynomial in terms of the basis vectors. In this case, the basis is given as p1=1+x, p2=1-x, p3=1+2x+3x^2. To find the coordinate vector of p(x)=1+x+x^2, we can express the polynomial as a linear combination of the basis vectors:

p(x)=1(1+x)+1(1-x)+0(1+2x+3x^2)

Therefore, the coordinate vector of p(x) relative to the given basis is [1, 1, 0].

2) To find an orthonormal basis for X, we can use the Gram-Schmidt process. This process takes a set of vectors and produces an orthonormal set of vectors with the same span.

In this case, we have the vectors (1,1,1,1), (1,1,1,0), and (1,1,0,0). First, we normalize the first vector by dividing it by its length:

u1=(1,1,1,1)/√4=1/2(1,1,1,1)

Next, we subtract the projection of the second vector onto u1 from the second vector:

v2=(1,1,1,0)-proj((1,1,1,0),u1)=(1,1,1,0)-1/2(1,1,1,1)=(1/2,1/2,1/2,-1/2)

Then, we normalize v2 to obtain u2:

u2=(1/2,1/2,1/2,-1/2)/√3=1/6(1,1,1,-1)

Finally, we repeat the process for the third vector:

v3=(1,1,0,0)-proj((1,1,0,0),u1)-proj((1,1,0,0),u2)=(1,1,0,0)-1/2(1,1,1,1)-1/6(1,1,1,-1)=(1/3,1/3,-1/6,-1/6)

Normalizing v
 

1. What is orthogonal projection?

Orthogonal projection is a mathematical process that involves finding the closest vector in a subspace to a given vector. It is commonly used in linear algebra and vector calculus to solve problems related to linear transformations.

2. What is an orthonormal basis?

An orthonormal basis is a set of vectors in a vector space that are both orthogonal (perpendicular) and normalized (have a length of 1). This type of basis is useful because it simplifies calculations and allows for easier visualization of vector operations.

3. How is the coordinate vector of a polynomial determined?

The coordinate vector of a polynomial is determined by finding its coefficients with respect to a given orthonormal basis. This is usually done using the Gram-Schmidt process, which involves finding the orthogonal projection of the polynomial onto each vector in the basis and then normalizing the resulting vectors to obtain a set of orthonormal vectors.

4. How is orthogonal projection used in polynomial regression?

In polynomial regression, orthogonal projection is used to find the best-fit polynomial curve for a given set of data points. The orthogonal projection of the data points onto the polynomial basis gives the coefficients of the polynomial that minimizes the sum of squared errors between the data and the curve.

5. What are some real-world applications of orthogonal projection and orthonormal bases?

Orthogonal projection and orthonormal bases have a wide range of applications in fields such as engineering, physics, and computer graphics. Some examples include signal processing, image compression, and pattern recognition. They are also used in solving optimization problems and in the construction of efficient algorithms.

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