Linear Algebra: Projection onto a subspace

In summary, the conversation is about a homework problem involving a projection formula. The person attempted to solve it but got an incorrect answer. They question whether they need to use a specific number format and discuss alternative approaches. It is suggested that the issue may be that the vectors are not orthogonal.
  • #1
Kisa30
4
0

Homework Statement



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That is the question. The answer on the bottom is incorrect

Homework Equations



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I believe that is the formula that is supposed to be used.

The Attempt at a Solution



All I really did was plug in the equation into the formula but there is something I am missing because the answer is incorrect

Projection = (41/65)v1 + (26/5)v2
This is what I got after inserting the projection formula.
And in the first image, on the bottom it shows the final solutions I got.


Please help me figure out how to do this question and where I went wrong.

Thanks in advanced!
 
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  • #2
Just a thought, but do you have to provide the solutions in a specific number format (i.e. rounded to a certain number of figures) or maybe as exact fractions?
 
  • #3
I don't think it's important, no. =)
 
  • #4
What conditions must v1 and v2 meet so that the formula can be used?

An alternative approach would be to find a vector x that's perpendicular to V, and find the projection of v onto x, and subtract that from v. What's left over will lie in the subspace V.
 
  • #5
Apparently the problem is that it's not orthogonal.
 

FAQ: Linear Algebra: Projection onto a subspace

1. What is projection onto a subspace in linear algebra?

Projection onto a subspace in linear algebra is a mathematical operation that involves finding the closest point in a subspace to a given vector. It is used to decompose a vector into two components, one that lies in the subspace and one that is orthogonal to it.

2. Why is projection onto a subspace important?

Projection onto a subspace is important because it allows us to simplify complex vector problems by breaking them down into smaller, more manageable parts. It also has many applications in fields such as computer graphics, signal processing, and data analysis.

3. How do you calculate projection onto a subspace?

The formula for calculating projection onto a subspace involves finding the dot product between the given vector and a basis for the subspace, and dividing that by the dot product between the basis and itself. This will give you the coefficient for the basis vector, which can then be multiplied by the basis vector to get the projection.

4. What is the difference between projection onto a subspace and projection onto a vector?

Projection onto a subspace involves finding the closest point to a given vector within an entire subspace, while projection onto a vector involves finding the closest point to a given vector within a specific direction. In other words, projection onto a vector is a special case of projection onto a subspace.

5. How is projection onto a subspace used in real-world applications?

Projection onto a subspace has many practical applications, such as image and signal compression, data dimensionality reduction, and solving optimization problems. It is also used in machine learning algorithms, such as principal component analysis, which involves finding the projection of data onto a lower-dimensional subspace.

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