Linear Algebra Problem - Distance from the Image of a Matrix to a Vector

[Tangentiality]

Homework Statement

A problem on a linear algebra assignment is really confusing me. The problem is asking me to compute the distance from a vector to the image of a matrix (with orthogonal columns).
matrix A =
1 -2
4 6
2 -11

y=
3
-5
2

Homework Equations

dist(f,g) = norm( f - g ) ? possibly

The Attempt at a Solution

In the context of the actual course, the place I began was inner product spaces, seeing as that's really the only place where a distance computation is defined in the text. This is defined as dist(f,g) = norm( f - g ), where f and g are two elements of an inner product space. However, I wasn't sure where to go from there, or what operator I would use. I considered thinking about the matrix A as a plane and finding the normal vector, and computing a distance that way, but that seemed like a dead end as well. Any help would be appreciated.

 

[radou]

Distance to the image of an orthogonal matrix? First of all, this even isn't a square matrix, unless I'm missing something here?

 

[Tangentiality]

Distance to the image of an orthogonal matrix? First of all, this even isn't a square matrix, unless I'm missing something here?

I meant to say that the matrix in question has orthogonal columns (which is mentioned in the problem). I overlooked the fact that an orthogonal matrix is something completely different.

 

[radou]

OK, what "image" of your matrix do you mean?

 

[Tangentiality]

OK, what "image" of your matrix do you mean?

I'm assuming that the image in this case just implies the span of the column vectors of A.

 

[radou]

Well, you can find the general form of the vector v which is included in the span of the column vectors of A, in terms of some coefficients α and β, and then calculate d(y, b) = ||y - b||, if that's what you're trying to do.

 

[Tangentiality]

Let me see if I'm interpreting that correctly. That would involve solving a system of linear equations involving the columns of A and the components of the vector v. Then would I use that vector to compute the distance in terms of the inner product space? I'm a little unclear what b is in that equation.

 

[radou]

Oops, my mistake, what I meant was d(y, v) = ||y - v||. y is given, and v can be written in the form α(-1, 2, -2) + β(2, 2, 1) = (2β - α, 2α + 2β, -2α + β).

 

[lanedance]

or if i read it right you could save time by thinking geomtrically

the image space of the matrix is a 2D plane through the origin, spanned by the column vectors.

The shortest vector from the tip of the y to the plane, will be perpindicular to the plane...

 

[Tangentiality]

Thanks for your suggestions. I'm still pretty confused as to what exactly this problem wants me to find, but this definitely gives me some place to start.

 

[lanedance]

can you write the question exactly as written? its always agood start on here, just to make sure you haven't missed anything...

though from what you've written, I think it means find the shortest distance from the tip of the y to the plane. The shortest distance will always be perpidicular to the plane.

As both the plane and y pass through the origin. Now imagine the unique vector, call it p, perpindicular to the plane, that goes from the plane to the tip of y.

The length of the vector p is the distance you want to find.

HINT: think about the projection of y onto the perpindicular direction