Linear Algebra; Transformation of cross product

[Myr73]

oh ok, so how would I write out when v does not equal zero,that the range of T is the plane that is orthogonal to v

 

[Fredrik]

First of all, there's more than one such plane. Do you know which one of them is the range of T?

 

[Myr73]

v• (vXu)=0 , v is orthogonal to vXu

 

[Fredrik]

Yes, v is orthogonal to v×u for all u. But that doesn't answer the question. Which one of the infinitely many planes that are orthogonal to v is the range of T? You can answer this one in words.

By the way, if you click the Σ symbol in the green bar at the top of the editor area, you will see a list of symbols that you can use when you type. No need to type X instead of × for example.

 

[Myr73]

Oh, ok,cool thanks. I will use the symbol bars..
And , I do not know.. I really do not know.

 

[Fredrik]

Are you aware that the range of a linear operator is a subspace? You should verify that for yourself, if you haven't done it recently. It's a nice exercise. Now, of all the planes that are orthogonal to v, which ones are subspaces and which ones are not?

 

[Myr73]

yes, I am aware of that. and I know that to find a subspace of R3 that is orthogonal to v, we would find the orthogonal projection of R3 on W. where w is a subspace of R3. witch would be the Ax, A being the column space of given vectors of v, and x being the least squares solution of Ax=v.

 

[Fredrik]

So you know that

(1) The range of T is a plane orthogonal to v.
(2) The range of T is a subspace of V.

Now the question is, which of the planes that are orthogonal to v are subspaces? Think about what a subspace is.

 

[Myr73]

Ok, well i have answered a and b to the best of my knowledge and ability, is there someone that can help me on part c)?
Here is the question again.
1- Question
Let V be a fixed vector in R^3. a)Show that the transformation defined by T( u)= v X u is a linear transformation.
b) Find the range ot the linear transformation
c) If v=i , find the matrix for this linear transformation.

2-I do not know u in this equation, I figured to just take it as u=(u1,u2,u3), I know that v would equal v=(1,0,0). And that T(U)= v×u={v2u3-v3u2,v3u1-v1u3,v1u2-v2u1}, and that
[T]=[T(e1)|T(e2)|T(e3)].
From the first formula, I found it to be ( 0,-u3,u2). However I do not think it is correct.

Thank you

 

[Fredrik]

2-I do not know u in this equation,

What equation? If you mean the definition of T, that's not a statement about a vector called u. It's a statement about all vectors.

And that T(U)= v×u={v2u3-v3u2,v3u1-v1u3,v1u2-v2u1}

Just some advice about the notation: Don't type U when you mean u. These are two different symbols, so they don't automatically represent the same thing.

[T]=[T(e1)|T(e2)|T(e3)].
From the first formula, I found it to be ( 0,-u3,u2). However I do not think it is correct.

It looks like you ignored the formula that tells you what [T] is, and instead calculated T(u) for an arbitrary u.

 

[Myr73]

yes, I am sorry , a couple of minor errors there. I meant "u" not "U".

And yes I ignored the formula with [T], because I did not know what to do with it. But I thought they wanted to know if V=i then find T(u), but I guess not. How am I suppose to compute [T] if v=(1,0,0)?

 

[Fredrik]

The formula tells you that $Te_1$, $Te_2$ and $Te_3$ are the columns of [T]. You should probably take a look at the FAQ post I linked to early in this thread.

 

[Myr73]

Does [T]= 1 0 0
0 0 0
0 0 0 , make sense? And I will re-read it, I do not really understand much of it, but I will reread it.

 

[Fredrik]

It's not the right answer. For example, we have $Te_1=e_1\times e_1=0$.

If you ask about the things you find difficult to understand in the FAQ post, I will answer.

 

[micromass]

So Fredrik already gave you $Te_1$. What is $Te_2$ and $Te_3$?

 

[Myr73]

"If X=Y, it's convenient to choose B=A, and to speak of the matrix representation of T with respect to A instead of with respect to (A,A), or (A,B). The formula for Tij can now be written as
Tij=(Tej)i=(Tei),(Tej)"
I guess what I did not understand is how to refer it to the matrix of the transformation T(u)=v×u.

But by [T1]= (e1)×(e1)=0, would this be like doing the cross product of (1,0,0) and (1,0,0). ?
[T2]= (e2)×(e2)=0, [T3]=(e3)×(e3)=0. No that cannot be it would simply give a zero transformation matrix.

 

[micromass]

What is $Te_2$ and $Te_3$?

 

[Fredrik]

"If X=Y, it's convenient to choose B=A, and to speak of the matrix representation of T with respect to A instead of with respect to (A,A), or (A,B). The formula for Tij can now be written as
Tij=(Tej)i=(Tei),(Tej)"
I guess what I did not understand is how to refer it to the matrix of the transformation T(u)=v×u.

You posted the formula $[T]=[Te_1|Te_2|Te_3]$, and I explained what it means. Why not use that?

The formula in the FAQ post is saying the same thing: The number on row i, column j of the matrix [T] is denoted by $[T]_{ij}$. The formula from the FAQ says that $[T]_{ij}=(Te_j)_i$. That's the $i$th component of the vector $Te_j$. This formula tells you that the first column of $[T]$ is
$$\begin{pmatrix}[T]_{11}\\ [T]_{21}\\ [T]_{31} \end{pmatrix} =\begin{pmatrix}(Te_1)_1\\ (Te_1)_2\\ (Te_1)_3\end{pmatrix}.$$ This is exactly what the formula you posted says.

What you need to understand about the ordered bases mentioned at the start of the quote is this: There isn't just a single matrix associated with a linear operator $T:X\to Y$. There's one for each pair (E,F) such that E is an ordered basis for X and F is an ordered basis for Y. When someone mentions the matrix corresponding to a linear operator $T:\mathbb R^n\to\mathbb R^n$ and doesn't mention a pair (E,F) of ordered bases, you can assume that what they have in mind is that E = F = the standard ordered basis for $\mathbb R^n$.

The problem you're working on is missing a vital piece of information. Since $T:V\to V$, where V is an arbitrary finite-dimensional vector space, an ordered basis must be provided for the question to make sense. Since the problem doesn't mention an ordered basis, I think the only way to proceed is to assume that $V=\mathbb R^n$.

But by [T1]= (e1)×(e1)=0, would this be like doing the cross product of (1,0,0) and (1,0,0). ?

Yes, because $e_1=(1,0,0)$.

[T2]= (e2)×(e2)=0, [T3]=(e3)×(e3)=0.

It looks like you're just guessing now. You have to use the definition of T.

 

[Myr73]

agh ok, and no I was not guessing, the cross product of (0,1,0) and (0,1,0) is also zero...

 

[Myr73]

I am not sure, but i thought
T2=(e2)x(e2), and i thought e2 would be (0,1,0)

 

[Fredrik]

I am not sure, but i thought
T2=(e2)x(e2)

Why would you think that?

 

[Myr73]

because you said that [T1]=e1xe1

 

[Fredrik]

because you said that [T1]=e1xe1

Yes, but that's not a reason to think that the second column is $e_2\times e_2$. That's why I said that you seem to be guessing now.

You posted a formula that says that the second column is $Te_2$. Do you understand what this notation means?

 

[Myr73]

nvm, i have a final tomorrow - i will simply send it in as is, cause i obviously do not understand- and hope its not found to often on the exam! Thank you for your help -

 

[Fredrik]

In that case, I can only give you some general advice:

• Make sure that you understand the term function very well. Another word for the same thing is map. In particular, you need to know this: If f is a function and x is an element of its domain, then f(x) is an element of the codomain called the value of f at x. To define a function f is to say what its domain is, and then say what f(x) is for each x in the domain.
• If you've been given a definition of a specific function f, and you're asked to determine what f(x) is for some specific x in the domain, you cannot possibly find a better way to do this than to just use the definition of f. In particular, you can't conclude that f(3)=9 just because you know that f(2)=4.
  
• Make sure that you understand what it means for a function from a vector space into a vector space to be linear. A linear function from a vector space into a vector space is called a linear transformation.
  
• If T is a linear transformation and x is an element of its domain, it's standard to denote the value of T at x by Tx rather than T(x).
• Make sure that you understand the concepts linearly independent, span and basis. A basis for a vector space V is a linearly independent subset of V that spans V.
• An ordered basis for an n-dimensional vector space V is an n-tuple $(v_1,\dots,v_n)$ such that $\{v_1,\dots,v_n\}$ is a basis for V.
• There's a basis for $\mathbb R^n$ that's considered "standard". The standard basis vectors are denoted by $e_1,\dots,e_n$. They are defined by $e_1=(1,0,\dots,0)$, $e_2=(0,1,0,\dots,0)$, ..., $e_n=(0,\dots,0,1)$. The standard ordered basis for $\mathbb R^n$ is $(e_1,\dots,e_n)$.
• If x is an element of an n-dimensional vector space V, and $(v_1,\dots,v_n)$ is an ordered basis for V, then there's exactly one n-tuple $(x_1,\dots,x_n)$ of elements of V such that $x=\sum_{i=1}^n x_i v_i$. The components of this n-tuple (i.e. the vectors $x_1,\dots,x_n$) are called the components of v with respect to this particular ordered basis.
  
• There isn't just one matrix associated with a linear transformation $T:X\to Y$. There's one for each pair $(E,F)$ such that E is an ordered basis for X and F is an ordered basis for Y.
  
• The formula for the matrix $[T]_{F,E}$ of a linear transformation $T:X\to Y$ with respect to a pair $(E,F)$ such that E is an ordered basis for X and F is an ordered basis for Y, is $([T]_{F,E})_{ij}=(Te_j)_i$, where $e_j$ is the $j$ the component of the ordered basis $E$ and $(Te_j)_i$ is the $i$th component of $Te_j$ with respect to F. This formula implies that the $j$th column of $[T]_{F,E}$ consists of the components of $Te_j$ with respect to $F$.
• When it's clear from the context which pair of ordered bases is involved, the notation can be simplified from $[T]_{F,E}$ to just $[T]$.
  
• When we're dealing with a linear transformation $T$ whose domain and codomain are both $\mathbb R^n$, the matrix of $T$ with respect to $(E,E)$, where E is the standard ordered basis for $\mathbb R^n$, is often referred to as "the" matrix of $T$.
  
• These things are important. If you're going to study math up to the level of differential geometry or physics up to the level of quantum mechanics, you should make sure that you understand these things perfectly.