contestada

Find the matrices of the following linear transformations
from R3 to R3.
The orthogonal projection onto the xy-plaNE: (3X3)...
The reflection in the z-axis:(3X3)

Respuesta :

ogorwyne

Answer:

The matrixes are in the explanation box.

1. A mapping of (x,y,z) to (x,y,0)

2. Is a mapping of (x,y,z) to (-x,-y,z)

Step-by-step explanation:

We have the orthogonal projection onto the cy-plane to be a mapping of (x,y,z) to (x,y,0). Then we will have a transformation matrix that has the form below.

[1 0 0]

[0 1 0]

[0 0 1]

For the reflection in the z-axis,it is a reflection of (x,y,z) to (-x,-y,z). Our matrix will then have the form below

[-1 0 0]

[0 -1 0]

[0 0  1]

Ver imagen ogorwyne
facundo3141592

The matrix that projects into the XY plane is:

[tex]\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&0\end{array}\right][/tex]

The matrix that reflects in the z-axis is:

[tex]\left[\begin{array}{ccc}-1&0&0\\0&-1&0\\0&0&1\end{array}\right][/tex]

How to find the transformations?

First, we want to find the transformation that goes from (x, y, z) to (x, y, 0).

Remember that:

[tex](x, y, z)*\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right] = (x, y, z)[/tex]

That matrix is the identity matrix.

Then, if we want to remove the z component, then we just need to make the last row-column equal to zero, we will get:

[tex](x, y, z)*\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&0\end{array}\right] = (x, y, 0)[/tex]

Now we want to make a reflection over the z-axis, this is a rotation of 180° around the z-axis, so it changes the sign of the x and y components:

(x, y, z) to (-x, -y, z)

From this, we conclude that the matrix will be:

[tex](x, y, z)*\left[\begin{array}{ccc}-1&0&0\\0&-1&0\\0&0&1\end{array}\right] = (-x, -y, z)[/tex]

If you want to learn more about matrices, you can read:

https://brainly.com/question/11989522

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