Hi Guys,

I have these linear transformation problems which have caused me some trouble today.

I hope You can help me.

a) [tex](x,y) \rightarrow (x+3,y+5)[/tex] is called a linear translation according to my Linear Algebra textbook.

I'm tasked with showing that the above can't be done as linear transformation by using regular coordinates. Secondly I'm tasked with providing a linear transformation using homogeneous coordinants, which does the translation.

Any hits idears on how I do that??

I know what the translation can be written as

[tex]\left[ \begin{array}{ccc} 1 & 0 & 3 \\ 0 & 1 & 5 \\ 0 & 0 & 1 \end{array} \right]\left[ \begin{array}{c} x \\ y \\ 1 \end{array} \right] = \left[ \begin{array}{c} x + 3 \\ y+5 \\ 1 \end{array} \right][/tex]

But does that help me in any way proving the above ??

b) I'm tasked provinding a linear transformation which rotates the following rectangle including edges 90 degress clockwise around the center of the figure.

I'm provided the following coordinants for the figure.

A(3,1) B(5/2, 2), C(9/2) and D(5,2).

In hits ideers on how I do this ?

Finally C)

Show that the following linear transformation t(x) is given in regular coordinants,

[tex] T(x) = \left[ \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right] \cdot x[/tex]

such that it can be completed by this linear transformation

[tex] T(x) = \left[ \begin{array}{ccc} a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & 0 \\ 0 & 0 & 1 \end{array} \right] \cdot x[/tex]

using homogeneous coordinants.

In hints or ideer on how I do that?

Sincerely and best Regards,

Fred

I have these linear transformation problems which have caused me some trouble today.

I hope You can help me.

a) [tex](x,y) \rightarrow (x+3,y+5)[/tex] is called a linear translation according to my Linear Algebra textbook.

I'm tasked with showing that the above can't be done as linear transformation by using regular coordinates. Secondly I'm tasked with providing a linear transformation using homogeneous coordinants, which does the translation.

Any hits idears on how I do that??

I know what the translation can be written as

[tex]\left[ \begin{array}{ccc} 1 & 0 & 3 \\ 0 & 1 & 5 \\ 0 & 0 & 1 \end{array} \right]\left[ \begin{array}{c} x \\ y \\ 1 \end{array} \right] = \left[ \begin{array}{c} x + 3 \\ y+5 \\ 1 \end{array} \right][/tex]

But does that help me in any way proving the above ??

b) I'm tasked provinding a linear transformation which rotates the following rectangle including edges 90 degress clockwise around the center of the figure.

I'm provided the following coordinants for the figure.

A(3,1) B(5/2, 2), C(9/2) and D(5,2).

In hits ideers on how I do this ?

Finally C)

Show that the following linear transformation t(x) is given in regular coordinants,

[tex] T(x) = \left[ \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \end{array} \right] \cdot x[/tex]

such that it can be completed by this linear transformation

[tex] T(x) = \left[ \begin{array}{ccc} a_{11} & a_{12} & 0 \\ a_{21} & a_{22} & 0 \\ 0 & 0 & 1 \end{array} \right] \cdot x[/tex]

using homogeneous coordinants.

In hints or ideer on how I do that?

Sincerely and best Regards,

Fred

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