# Image of a Linear Transformation

• pondzo
In summary, T2 projects orthogonally onto the xz-plane, T3 rotates clockwise through an angle of 3π/4 radians about the x axis, and the point (-3, -4, -3) is first mapped by T2 and then T3. what are the coordinates of the resulting point? this question is on a program call Calmaeth. My answer for this question is (-3,0,-√2/2). The program says its wrong but i have checked thoroughly many times and cannot find my mistake. My transformation matrix for T2 is ##\begin{pmatrix}1 & 0 & 0 \\0 & 0 & 0 \\0 & 0
pondzo
T2 projects orthogonally onto the xz-plane

T3 rotates clockwise through an angle of 3π/4 radians about the x axis

The point (-3, -4, -3) is first mapped by T2 and then T3. what are the coordinates of the resulting point?

this question is on a program call Calmaeth. My answer for this question is (-3,0,-√2/2). The program says its wrong but i have checked thoroughly many times and cannot find my mistake.

My transformation matrix for T2 is ##
\begin{pmatrix}
1 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 1
\end{pmatrix}
## and for T3 is ##
\begin{pmatrix}
1 & 0 & 0 \\
0 & \frac{-1}{√2} & \frac{1}{√2} \\
0 & \frac{-1}{√2} & \frac{-1}{√2}
\end{pmatrix}
##
To get the resulting standard matrix, i did T3*T2 and then multiplied this matrix by the point (-3, -4, -3) to get the resulting point.

Can anyone see where i went wrong if i did? (also to let you know, the program said my matrices for T2&T3 were correct)

Your method is fine. Assuming your matrices are right, it looks like you made an error with the matrix multiplication. What did you get for ##T3 * T2##?

I agree with jbuniii. The "-3" and "0" are correct but I do not get $-\sqrt{2}/2$ as the third component when I multiply your matrices.

(As a check, rather than multiplying T3*T2 first and then multiplying that by the vector, you can multiply the vector by T2 and then multiply the result by T3.)

Last edited by a moderator:
HallsofIvy said:
The "-3" and "0" are correct
How is the "0" correct?

oay said:
How is the "0" correct?

Yes, only the ##-3## is correct.

Anyway, please post this in the homework forum next time! Thanks

Hi guys It was a rookie mistake on my part. I was doing [-3,-4,-3]*[standard matrix] . when i should have been doing [standard matrix]*##\begin{pmatrix}
-3 \\
-4 \\
-3
\end{pmatrix}##
And sorry, in the future ill post in the homework section.

## 1. What is an image of a linear transformation?

An image of a linear transformation refers to the set of all possible outputs that can be obtained from applying the transformation to a given set of inputs. It is also known as the range of the transformation.

## 2. How is the image of a linear transformation different from the preimage?

The preimage of a linear transformation refers to the set of all inputs that will produce a given output when the transformation is applied. The image, on the other hand, refers to the set of all outputs that can be obtained from applying the transformation to a given set of inputs. In other words, the preimage is the input while the image is the output.

## 3. Can the image of a linear transformation be larger than the preimage?

Yes, it is possible for the image of a linear transformation to be larger than the preimage. This can happen when the transformation maps multiple inputs to the same output, resulting in a larger range of possible outputs.

## 4. How is the image of a linear transformation related to the null space?

The image of a linear transformation and the null space are complementary subspaces. This means that the dimensions of the image and null space combined are equal to the dimension of the original vector space. Additionally, any vector in the null space will be mapped to the zero vector in the image, and vice versa.

## 5. How can the image of a linear transformation be visualized?

The image of a linear transformation can be visualized as the transformation of a geometric shape, such as a square or a circle. The resulting shape after applying the transformation will be the image. For higher dimensions, the image can be represented as a set of points in space.

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