Projection onto Column Space of A and its Perpendicular

In summary, The question involves finding the projection of a vector onto the column space of a matrix. The steps to do this include finding the column space of the matrix, which can be done by taking the reduced row echelon form and shrinking it if necessary. The perpendicular of a matrix can be found by using the definition that v is perpendicular to a space if v dot u is equal to 0 for all u in that space. Further explanation may be needed for finding the perpendicular of a matrix.
  • #1
Zyxer22
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Homework Statement



Some of the details in this question are based off the use of matlab. If it's needed I can show the matrices that MATLAB creates.


Let A = magic(8); A = A(:,1:3) and let S be the Column Space of A. For b = [1:8]' compute the projection of b onto the Column Space of A. What is the projection of b perpendicular to the Column Space of A?



Homework Equations





The Attempt at a Solution



I'm not sure exactly where to get started. My first issue is that I'm not sure how to find the column space of A. I think I'm supposed to take the reduced row echelon form of the matrix and then shrink it if necessary. My book defines Column space as the span, so this would make sense to me.

I'm also not sure how to find the perpendicular of a matrix. I know it's defined such that v \in W^\perp => v \cdot u = 0 | u \in W but I'm not sure how to go from there.
 
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  • #2
I'm fairly sure I'm right about the column space but I could use an explanation on perpendicular matrices if anyone can help.
 

Related to Projection onto Column Space of A and its Perpendicular

1. What is a projection in linear algebra?

A projection in linear algebra is a mathematical operation that maps a vector onto a lower-dimensional subspace. This can be thought of as "projecting" the vector onto a specific plane or line in the vector space.

2. How is a projection represented in a matrix?

A projection can be represented by a matrix called a projection matrix. This matrix is typically a square matrix that, when multiplied by a vector, produces the projection of that vector onto the subspace.

3. What is the purpose of projections in linear algebra?

Projections are used in linear algebra to simplify calculations and solve problems involving vectors and subspaces. They also have applications in geometry, physics, and computer graphics.

4. What are the properties of a projection matrix?

A projection matrix is idempotent, meaning that when multiplied by itself, it produces the same matrix. It is also symmetric and has eigenvalues of 0 or 1. Additionally, the column space of a projection matrix is equal to the subspace it is projecting onto.

5. How do projections relate to orthogonal vectors?

If a vector is projected onto an orthogonal subspace, the resulting projection vector will be orthogonal to the original vector. This means that the dot product of the original vector and the projection vector will be 0. Projections can also be used to find the orthogonal complement of a subspace.

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