Matrix of a linear transformation

[succubus]

Find a basis of Rn such that the matrix B of the given linear transformation T is diagonal.

Orthogonal Projection T onto the line in R^3 spanned by
(1 1 1)

I'm assuming (though I tend to be wrong) that I need to find a vector that is parallel to the line and 2 that are perpendicular to it and linearly independent.

So would the B matrix look something like this

1 -2 1

1 1 0

1 1 -1

?

Or am I once again way off track :)

 

[lanedance]

Hi Succubus
misread question... run with mazes comments

 

[maze]

Find a basis of Rn such that the matrix B of the given linear transformation T is diagonal.

Orthogonal Projection T onto the line in R^3 spanned by
(1 1 1)

I'm assuming (though I tend to be wrong) that I need to find a vector that is parallel to the line and 2 that are perpendicular to it and linearly independent.

So would the B matrix look something like this

1 -2 1

1 1 0

1 1 -1

?

Or am I once again way off track :)

Looks good. A few follow up questions worth thinking about would be,
1) If you write a vector in the basis B, v = v₁ (1 1 1)^T + v₂ (-2 1 1)^T + v₃ (1 0 -1)^T, then what is Tv in this basis?
2) What is the matrix T in the basis B?
3) What would go wrong if the second vector in B was not perpendicular to the first one?

*extra challenge*4) What is T in the standard basis (1,0,0)^T, (0,1,0)^T, (0,0,1)^T? Can you write the projection matrix in terms of an outer product (ie: T = q q^T for some vector q)?

 

[succubus]

Thanks, I appreciate your help. Now, what is it was a reflection instead of a projection? Let's say a reflection about the plane (1 1 1) Would I do the same procedure?

 

[lanedance]

hi succubus - do you mean the reflection in the plane through the origin, with normal vector (1,1,1) ?

if so, you can look at your previous vectors (2 perpindicular and 1 parallel to (1,1,1)).

Say v₁= (1,1,1), this will be mapped to (-1,-1,-1). Whilst the perpindicular vectors (v₂,v₃) will be mapped to themselves,as they are in the plane of reflection

so you know
$$A.v_1 = -v_1$$
$$A.v_2 = v_2$$
$$A.v_3 = v_3$$

can you write A in the basis of v₁,v₂,v₃? then you can you transform to the original basis if needed...

 

[succubus]

Thanks! But the problem is

Find a basis of Rn such that the matrix B of the given linear transformation T is diagonal.

Reflection T about the line in R^3 spanned by
(1 1 1)


I accidentally put plane! I'm sorry.

So I would find 2 vectors perpendicular to the line and 1 parallel? I think I see it, but I don't understand exactly why. :/

So it would be

1 -1 1
1 0 -1
1 1 0

?

 

[lanedance]

hmm.. ok, so i assume it will reflect any vector u, in a plane parallel to (1,1,1)
and if we write
$$u = v+ \lambda(1,1,1)$$ for some unqiue $$v, \lambda$$ for every u
v will be normal to the plane of reflection...(someone correct me if i am wrong)

as before, say you have vectors
v₁ - parallel (1,1,1)
v₂ - perp (1,1,1)
v₃ - perp (1,1,1) (v₃ not equal v₂)

in this case
$$A.v_1 = v_1$$
$$A.v_2 = -v_2$$
$$A.v_3 = -v_3$$

then comments as before