# Projections onto subspaces

• shaon0
In summary, the conversation discusses finding the projection of a vector onto a plane, with the use of the projection formula. It also mentions finding the matrix representing a linear mapping and provides a helpful resource for this topic. Finally, it suggests two different methods for finding the projection, one involving finding a basis and making it orthogonal, and the other involving resolving the vector into components parallel and perpendicular to the plane.

See attachment

## The Attempt at a Solution

How should I approach these questions? By using the projection formula?

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• Projections onto planes.png
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Yes, since the problem asks for projections, use the projection formula!

HallsofIvy said:
Yes, since the problem asks for projections, use the projection formula!

Will I have to do two different calculations ie one with v and the first vector in W and then v with the second vector in W? Also, how would I find P?

W is a plane. The question is asking you to find the projection of the vector v onto that plane.

As far as finding the matrix P goes, the projection onto W is a linear mapping. How do you find the matrix representing a linear mapping?

vela said:
W is a plane. The question is asking you to find the projection of the vector v onto that plane.

As far as finding the matrix P goes, the projection onto W is a linear mapping. How do you find the matrix representing a linear mapping?

For P, the projection of what vector onto W? Would I just span the vectors I've found so, P=span{v1,v2} and find the co-effs s.t Basis1=a1.v1+a2.v2 where v1,v2 are the vectors I've found using the projection formula and a1,a2 are constants which will give me the 1st column of P?

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You might find the page helpful:

http://www.cliffsnotes.com/study_guide/Projection-onto-a-Subspace.topicArticleId-20807,articleId-20792.html [Broken]

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vela said:
You might find the page helpful:

http://www.cliffsnotes.com/study_guide/Projection-onto-a-Subspace.topicArticleId-20807,articleId-20792.html [Broken]
I've tried to use the projWv=projv1v+projv2v where v1=(-2,,1,-2)^T and v2=(1,4,-8)^T but i don't get the correct answer. Maybe because v1, v2 are not mutually orthogonal?

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Since any vector $v$ can be written as a linear combination of vectors of a basis of $\mathbb{R}^3$, if you can find the projection of each of the vectors of the canonical basis of $\mathbb{R}^3$, you can then write a projection matrix using the results.

shaon0 said:
I've tried to use the projWv=projv1v+projv2v where v1=(-2,,1,-2)^T and v2=(1,4,-8)^T but i don't get the correct answer. Maybe because v1, v2 are not mutually orthogonal?
Yes, that's why it didn't work.

unlearned said:
Since any vector $v$ can be written as a linear combination of vectors of a basis of $\mathbb{R}^3$, if you can find the projection of each of the vectors of the canonical basis of $\mathbb{R}^3$, you can then write a projection matrix using the results.

So; proje1v1? for all combinations of e1,e2,e3 and v1, v2?

If you look at the first picture on that web page, it illustrates that you can write
$$\vec{v} = (\mathrm{proj}_W\ \vec{v}) + \vec{v}_\perp$$That is, you can resolve any vector $\vec{v}$ into a piece that lies in the plane W and a piece that's perpendicular to W. Solving for the projection, you get
$$\mathrm{proj}_W\ \vec{v} = \vec{v} - \vec{v}_\perp$$So if you can figure out how to find $\vec{v}_\perp$, which is very likely a problem you solved before, you can then find the projection of $\vec{v}$ onto W. Hint: think about the normal to the plane.

If you don't want to use that approach, you can go with the method you tried. But as you noted, you need an orthogonal basis for W. You've been given a basis. You just need to make it orthogonal.

vela said:
If you look at the first picture on that web page, it illustrates that you can write
$$\vec{v} = (\mathrm{proj}_W\ \vec{v}) + \vec{v}_\perp$$That is, you can resolve any vector $\vec{v}$ into a piece that lies in the plane W and a piece that's perpendicular to W. Solving for the projection, you get
$$\mathrm{proj}_W\ \vec{v} = \vec{v} - \vec{v}_\perp$$So if you can figure out how to find $\vec{v}_\perp$, which is very likely a problem you solved before, you can then find the projection of $\vec{v}$ onto W. Hint: think about the normal to the plane.

If you don't want to use that approach, you can go with the method you tried. But as you noted, you need an orthogonal basis for W. You've been given a basis. You just need to make it orthogonal.

Ok, thanks. I think I've got it.

## 1. What are projections onto subspaces?

Projections onto subspaces are mathematical operations that involve taking a vector and finding its closest representation within a given subspace. This can be thought of as "projecting" the vector onto the subspace, hence the name.

## 2. How are projections onto subspaces calculated?

The calculation of projections onto subspaces involves finding the orthogonal projection of a vector onto a given subspace. This can be done using various methods such as the Gram-Schmidt process or using matrix multiplication.

## 3. What is the purpose of projections onto subspaces?

The purpose of projections onto subspaces is to simplify mathematical problems and calculations by reducing the dimensionality of the problem. It is also used in various applications such as data compression, signal processing, and machine learning.

## 4. Are projections onto subspaces reversible?

No, projections onto subspaces are not reversible. This is because information is lost during the projection process, making it impossible to reconstruct the original vector from its projection onto a subspace.

## 5. Can projections onto subspaces be applied to any vector space?

Yes, projections onto subspaces can be applied to any vector space. However, the subspace onto which the vector is being projected must be a subspace of the original vector space. In other words, the subspace must contain all possible linear combinations of its basis vectors.