Help Linear Algebra transformations, is my understanding correct?

[degs2k4]

Hello,

I would like to check if my understanding of this linear algebra problem dealing with transformations is correct:

Part (1)

we have the following coordinates systems:

$$\tilde{x} = \begin{pmatrix} x_1 \\ x_2 \\ 1 \end{pmatrix}$$ and $$x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$$

and the following equation : $$\tilde{x}^T \tilde{A} \tilde{x} = 0$$

This parts only asks for getting the A matrix (symmetric matrix with coefficients) from a quadratic curve. Taking a look at the final parts of the problem, I guess this part is some sort of coordinates change from $$\tilde{x} \rightarrow x$$ using a matrix A, is this right? (like some sort of 3D-2D transformation) Part (2)

we have the following coordinates systems:

$$x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$$ and $$y = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}$$

and the following equation : $$x = Py$$

My thoughts: The equation in (i) corresponds to a quadratic curve, which has been rotated (it has a cross product term) and translated (it has terms y1^2, y1). In this part we want to transform (i) into (ii). In this transformation we undo the rotation (by rotating) and the matrix used is for it is P, transforming x coords into y coords.


Part (3)

we have the following coordinates systems:

$$y = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}$$ and $$z = \begin{pmatrix} z_1 \\ z_2 \end{pmatrix}$$

My thoughts: The equation in (ii) corresponds to a quadratic curve, which has been translated (it only has terms y1^2, y1). In this part we want to transform (ii) into (iii). In this transformation we undo the translation (by translating), with an equation like this:

$$z = y + M$$

transforming coords y into coords z. Part (4.A)

we have the following coordinates systems:

$$z = \begin{pmatrix} z_1 \\ z_2 \end{pmatrix}$$ and $$\tilde{z} = \begin{pmatrix} z_1 \\ z_2 \\ 1 \end{pmatrix}$$

and the following equation : $$\tilde{z}^T \tilde{B} \tilde{z} = 0$$

This parts looks like part 1, getting the matrix from a quadratic curve. Taking a look at the rest of the parts of the problem, I guess this part is a coordinates change from $$z \rightarrow \tilde{z}$$ using a matrix B, is this right? Part (4.B)

we have the following coordinates systems:

$$\tilde{z} = \begin{pmatrix} z_1 \\ z_2 \\ 1 \end{pmatrix}$$ and $$\tilde{x} = \begin{pmatrix} x_1 \\ x_2 \\ 1 \end{pmatrix}$$

and the following equation : $$\tilde{x} = \tilde{Q} \tilde{z}$$

This part is the composite transformation from (i), using coords $$\tilde{x}$$ to (iii) using coords $$\tilde{z}$$. So composing all transformations from the previous parts into one, my guess is that :

$$\tilde{Q} = ((\tilde{A} P) + M)\tilde{B}$$

would this be ok?

Thanks in advance and sorry for the long post...

 

[HallsofIvy]

Hello,

I would like to check if my understanding of this linear algebra problem dealing with transformations is correct:

Part (1)

we have the following coordinates systems:

$$\tilde{x} = \begin{pmatrix} x_1 \\ x_2 \\ 1 \end{pmatrix}$$ and $$x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$$

and the following equation : $$\tilde{x}^T \tilde{A} \tilde{x} = 0$$

This parts only asks for getting the A matrix (symmetric matrix with coefficients) from a quadratic curve. Taking a look at the final parts of the problem, I guess this part is some sort of coordinates change from $$\tilde{x} \rightarrow x$$ using a matrix A, is this right? (like some sort of 3D-2D transformation)

You want
$$\begin{bmatrix}x_1 & x_2 & 1\end{bmatrix}\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}\begin{bmatrix}x_1 \\ x_2 \\ 1\end{bmatrix}= x_1^2+ x_2^2- 6x_1x_2+ 2x_1+ 4x_2+ \frac{15}{9}$$
That gives you a number of equations for a, b, c, d, e, f, g, h, and i.

Part (2)

we have the following coordinates systems:

$$x = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$$ and $$y = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}$$

and the following equation : $$x = Py$$

My thoughts: The equation in (i) corresponds to a quadratic curve, which has been rotated (it has a cross product term) and translated (it has terms y1^2, y1). In this part we want to transform (i) into (ii). In this transformation we undo the rotation (by rotating) and the matrix used is for it is P, transforming x coords into y coords.


Part (3)

we have the following coordinates systems:

$$y = \begin{pmatrix} y_1 \\ y_2 \end{pmatrix}$$ and $$z = \begin{pmatrix} z_1 \\ z_2 \end{pmatrix}$$

My thoughts: The equation in (ii) corresponds to a quadratic curve, which has been translated (it only has terms y1^2, y1). In this part we want to transform (ii) into (iii). In this transformation we undo the translation (by translating), with an equation like this:

$$z = y + M$$

transforming coords y into coords z.


Part (4.A)

we have the following coordinates systems:

$$z = \begin{pmatrix} z_1 \\ z_2 \end{pmatrix}$$ and $$\tilde{z} = \begin{pmatrix} z_1 \\ z_2 \\ 1 \end{pmatrix}$$

and the following equation : $$\tilde{z}^T \tilde{B} \tilde{z} = 0$$

This parts looks like part 1, getting the matrix from a quadratic curve. Taking a look at the rest of the parts of the problem, I guess this part is a coordinates change from $$z \rightarrow \tilde{z}$$ using a matrix B, is this right?


Part (4.B)

we have the following coordinates systems:

$$\tilde{z} = \begin{pmatrix} z_1 \\ z_2 \\ 1 \end{pmatrix}$$ and $$\tilde{x} = \begin{pmatrix} x_1 \\ x_2 \\ 1 \end{pmatrix}$$

and the following equation : $$\tilde{x} = \tilde{Q} \tilde{z}$$

This part is the composite transformation from (i), using coords $$\tilde{x}$$ to (iii) using coords $$\tilde{z}$$. So composing all transformations from the previous parts into one, my guess is that :

$$\tilde{Q} = ((\tilde{A} P) + M)\tilde{B}$$

would this be ok?

Thanks in advance and sorry for the long post...

 

[degs2k4]

You want
$$\begin{bmatrix}x_1 & x_2 & 1\end{bmatrix}\begin{bmatrix}a & b & c \\ d & e & f \\ g & h & i\end{bmatrix}\begin{bmatrix}x_1 \\ x_2 \\ 1\end{bmatrix}= x_1^2+ x_2^2- 6x_1x_2+ 2x_1+ 4x_2+ \frac{15}{9}$$
That gives you a number of equations for a, b, c, d, e, f, g, h, and i.

Thanks for your reply!

Yes... I already got the values for a-i. What are you trying to say ?