Linear Algebra HELP: Closest Vector to v in R^4

In summary, in Euclidean space R^4 equipped with Euclidean Inner Product, the vector in subspace W that is closest to v=(1,0,0,-1) is the orthogonal projection of v into W. This can be found by differentiating the squared distance vector between an arbitrary element in W and v and setting the results equal to zero. By solving the resulting system of equations, the minimizing values of \alpha,\beta,\gamma can be found, giving the vector in W with the shortest distance to v.
  • #1
rayveldkamp
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Ok need help with a question that could be in tomorrow's exam:
In Euclidean space R^4 equipped with Euclidean Inner Product, let W be subspace which has basis {(1,0,-1,0), (0,-1,0,1), (2,1,-3,0)}.
Determine the vector in W which is closest to v=(1,0,0,-1)
 
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  • #2
That vector must be the orthogonal projection of v into W.
Proof:
Consider the squared distance vector between an arbitrary element in W and v:
[tex](v-\alpha{W}_{1}-\beta{W}_{2}-\gamma{W}_{3})^{2}[/tex]
This is minimized by differentiating with respect to [tex]\alpha,\beta,\gamma[/tex] and setting the three results equal to zero.
For example, we gain by differentiating with respect to [tex]\alpha[/tex]
[tex]-2(v-\alpha{W}_{1}-\beta{W}_{2}-\gamma{W}_{3})\cdot{W}_{1}=0[/tex]
That is, the distance vector is orthogonal to [tex]W_{1}[/tex]

By solving the system of equations, you gain the minimizing values of [tex]\alpha,\beta,\gamma[/tex] and hence, the vector in W with shortest distance to v.
[tex]W_{1},W_{2},W_{3}[/tex] is of course the basis of W.
 
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  • #3


To find the vector in W that is closest to v, we can use the orthogonal projection formula:

proj_w(v) = ((v · u1)/(u1 · u1))u1 + ((v · u2)/(u2 · u2))u2 + ((v · u3)/(u3 · u3))u3

Where u1, u2, and u3 are the basis vectors for W.

First, we need to find the dot products of v with each of the basis vectors:

v · u1 = (1,0,0,-1) · (1,0,-1,0) = 1
v · u2 = (1,0,0,-1) · (0,-1,0,1) = -1
v · u3 = (1,0,0,-1) · (2,1,-3,0) = 2

Next, we need to find the dot products of each basis vector with itself:

u1 · u1 = (1,0,-1,0) · (1,0,-1,0) = 2
u2 · u2 = (0,-1,0,1) · (0,-1,0,1) = 2
u3 · u3 = (2,1,-3,0) · (2,1,-3,0) = 14

Now we can plug these values into the projection formula:

proj_w(v) = (1/2)(1,0,-1,0) + (-1/2)(0,-1,0,1) + (2/14)(2,1,-3,0)
= (1/2, 0, -1/2, 0) + (0, 1/2, 0, -1/2) + (4/14, 2/14, -6/14, 0)
= (7/14, 1/2, -7/14, -1/2)

Therefore, the vector in W that is closest to v is (7/14, 1/2, -7/14, -1/2).

This can also be verified by finding the distance between v and each vector in W and choosing the vector with the smallest distance. The distance formula is given by:

d(v, w) = ||v - w|| =
 

FAQ: Linear Algebra HELP: Closest Vector to v in R^4

What is linear algebra and why is it important?

Linear algebra is a branch of mathematics that deals with systems of linear equations and their representations in vector spaces. It is used to solve problems related to geometry, physics, engineering, and many other fields. It is important because it provides a framework for understanding and solving complex problems in a systematic way.

What is the closest vector to v in R^4 and how is it calculated?

The closest vector to v in R^4 is the vector that minimizes the distance between v and any other vector in R^4. This distance is known as the Euclidean distance. It can be calculated using the formula d(v,w) = √(∑(vi-wi)^2) where vi and wi are the corresponding components of v and w respectively. The vector w that minimizes this distance is the closest vector to v in R^4.

What are some real-world applications of finding the closest vector in R^4?

Finding the closest vector in R^4 has many practical applications. For example, it can be used to find the shortest path between two points in a graph, to minimize errors in data compression, and to determine the best fit line for a set of data points. It is also used in fields such as computer graphics, machine learning, and signal processing.

How can linear algebra be used to solve problems related to the closest vector in R^4?

Linear algebra provides a set of tools and techniques that can be used to solve problems related to the closest vector in R^4. These include matrix operations, vector operations, and concepts such as orthogonality and projections. By utilizing these tools, it is possible to efficiently find the closest vector to v in R^4 and solve related problems.

Are there any resources available for learning more about linear algebra and the closest vector in R^4?

Yes, there are many resources available for learning more about linear algebra and the closest vector in R^4. These include textbooks, online courses, and tutorials. Some recommended resources include "Linear Algebra and Its Applications" by David C. Lay and "Introduction to Linear Algebra" by Gilbert Strang. Additionally, there are many online resources and forums where you can find practice problems and ask for help from experts in the field.

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