Matrix A for Linear Map T: R3→R3

In summary: Remember, when in doubt, always break down the transformation into smaller, more manageable parts. Good luck with your homework! In summary, the task is to determine the matrix A for a linear map T: R3→R3 which is defined by first mapping the vector u onto v×u, where v=(-9,2,9), and then reflecting it in the plane x=z (positively oriented ON-system). The determinant for A can also be determined. The first part of the transformation can be represented by the matrix [v×u], while the reflection in the plane x=z can be represented by the matrix S = \left( \begin{array}{ccc}0 & 0 &
  • #1
Hannisch
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Homework Statement


Determine the matrix A for the linear map T: R3R3 which is defined by that the vector u first is mapped on v×u, where v=(-9,2,9) and then reflected in the plane x=z (positively oriented ON-system). Also determine the determinant for A.

Homework Equations





The Attempt at a Solution


I actually started with the determinant and said that since the first mapping is a projection the determinant of that is =0 -- thus, the determinant for the whole thing is 0, since det(B*C)=det(B)*det(C) and in this case, A=S*P, where S is the determinant for the reflection (which is what we usually use in Swedish) and P is the projection.

Anyway, I started with S (for practise, if nothing else):

The plane will have the equation x-z=0 in its normal form and thus the normal to the plane is <1,0,-1>. So if I call vector w <a,b,c>, the reflection in the plane is:

<a,b,c> + t<1,0,-1> = <a+t,b,c-t>.

<a,b,c> + (t/2)<1,0,-1> needs to be on the plane and thus the coordinates for that vector need to fulfill the plane equation,

(a+t/2) - (c-t/2) = 0, t= -(a-c).

<a-(a-c),b,c+(a-c)>=<c,b,a>.

Thus, the reflection matrix is:
[tex]S = \left( \begin{array}{ccc}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0 \end{array} \right)\]
[/tex]


I didn't think that was too difficult, but now I'm entering the confusing part.

I can quite easily calculate v×u, if u=<x,y,z>.

(v×u = < -9y+2z, 9x+9z, -2x - 9y>.)

But (and I was thinking this from the very beginning) v×u is orthogonal to u (by definition of the cross product, because it creats a vector orthogonal to the plane containing v and u). But there can't be any projection if it's orthogonal, right? So I thought that if they all stay the same it should be the identity matrix and thus S*I=S, but that is not correct. And now I've got no idea what to do... And I didn't really want it to be the identity matrix, because that would not mean that the determinant is 0.

The conclusion is that I'm very confused.
 
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  • #2

Thank you for your post. I understand your confusion and would be happy to provide some guidance.

Firstly, let's address your concern about the projection. You are correct that the cross product of v and u will be orthogonal to both v and u. However, this does not mean that there is no projection happening. In fact, the cross product itself is a form of projection. It takes a vector and projects it onto a plane perpendicular to both v and u.

Now, to find the matrix A for this linear map, we need to consider the two parts of the transformation separately: the first mapping of u onto v×u, and then the reflection in the plane x=z.

For the first part, we can use the fact that the cross product can be represented by a matrix operation. This matrix is given by:

[v×u] = \left( \begin{array}{ccc}
0 & -v_z & v_y \\
v_z & 0 & -v_x \\
-v_y & v_x & 0 \end{array} \right)\]

where v_x, v_y, and v_z are the components of v.

So, the first part of the transformation can be represented by the matrix [v×u]. Now, for the reflection in the plane x=z, we can use a similar approach to what you did in your attempt. The normal to this plane is <1,0,-1>, so the reflection matrix would be:

S = \left( \begin{array}{ccc}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0 \end{array} \right)\]

Now, to find the matrix A for the entire transformation, we need to multiply these two matrices together. Remember that matrix multiplication is not commutative, so the order matters.

A = S[v×u] = \left( \begin{array}{ccc}
0 & 0 & 1 \\
0 & 1 & 0 \\
1 & 0 & 0 \end{array} \right)\] \left( \begin{array}{ccc}
0 & -v_z & v_y \\
v_z & 0 & -v_x \\
-v_y & v_x & 0 \end{array} \right)\]

Multiplying these two matrices together will give you the matrix A for the entire transformation.

I hope
 

FAQ: Matrix A for Linear Map T: R3→R3

1. What is a linear map T: R3→R3?

A linear map T: R3→R3 is a function that takes a vector in the 3-dimensional space R3 as an input and maps it to another vector in the same space. It is also called a linear transformation because it preserves the properties of linearity, such as scaling and addition, in the mapped vectors.

2. What is Matrix A for Linear Map T: R3→R3?

A matrix A for a linear map T: R3→R3 is a 3x3 matrix that represents the transformation of the input vector into the output vector. Each column in the matrix represents the coefficients of the linear combination of the input vector's coordinates that make up the corresponding coordinates of the output vector.

3. How is Matrix A calculated for a Linear Map T: R3→R3?

To calculate Matrix A for a linear map T: R3→R3, you can use the standard basis vectors of R3 as the input vectors and apply the linear transformation T to each basis vector. The resulting vectors will form the columns of Matrix A.

4. What is the significance of Matrix A in Linear Algebra?

Matrix A is significant in linear algebra because it allows for the representation and manipulation of linear transformations in a more efficient and organized manner. It also enables the use of matrix operations to solve systems of linear equations and perform other calculations in linear algebra.

5. How is Matrix A used to perform transformations in Linear Algebra?

In Linear Algebra, Matrix A can be multiplied with a vector to perform the linear transformation represented by the matrix. This is done by taking the dot product of each column of Matrix A with the vector, resulting in the mapped vector. Matrix A can also be used to perform other operations, such as finding the inverse of a linear transformation or determining the eigenvalues and eigenvectors of a transformation.

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