Illustrating Linear Transformation: Sketches for T

In summary, the conversation discusses how to draw sketches in order to illustrate that T is linear. It is important to keep the line L fixed while drawing vectors and applying T to them. The steps involve drawing vectors, adding them using the parallelogram law, and understanding the additivity and homogeneity of T. To approach part b, it is suggested to use trigonometry to express the components of T(x) in terms of x1, x2, and theta, and to find a matrix that represents this transformation. It is also noted that applying a matrix to the vector (1, 0) gives the first column of the matrix as a vector, and applying a matrix to the vector (0, 1) gives the
  • #1
yango_17
60
1

Homework Statement


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Homework Equations

The Attempt at a Solution


I would just like to know what is being requested when it asks me to draw sketches in order to illustrate that T is linear. Does it have something to do with altering to position of the line L itself? Any help would be very much appreciated. Thanks.
 
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  • #2
No, you keep the line ##L## fixed at all times in this exercise.
  1. Start by drawing a vector ##\mathbf{x}## in the diagram, say somewhere betwee the horizontal axis and the line ##L##. Now apply ##T## to ##\mathbf{x}##. Where does ##T(\mathbf{x})## go?
  2. After this, draw another vector ##\mathbf{y}## in the same way. Also draw ##T(\mathbf{y})## again.
  3. Next, add ##\mathbf{x}## and ##\mathbf{y}## using the parallelogram law. Call the result ##\mathbf{z}## and draw ##T(\mathbf{z})##.
  4. Finally, add ##T(\mathbf{x})## and ##T(\mathbf{y})## using the parallelogram law. If you have done the previous steps correctly, the resulting vector should coincide with ##T(\mathbf{z})##. Do you understand why this should be the case?
The previous steps illustrate the additivity of ##T##. It remains to sketch the homogeneity of ##T##. You can probably figure out how?
 
  • #3
Thank you! Any suggestions on how to approach part b?
 
  • #4
yango_17 said:
Any suggestions on how to approach part b?

Hint: In your sketch in step 1. above, using rules from trigonometry, you should be able to express ##[T(\mathbf{x})]_1## and ##[T(\mathbf{x})]_2## in terms of ##x_1, x_2## and ##\theta##. The ##2 \times 2## matrix
$$
A =
\begin{bmatrix}
a_{11} & a_{12}\\
a_{21} & a_{22}
\end{bmatrix}
$$
you are asked to find, is such that
$$
\begin{bmatrix}
[T(\mathbf{x})]_1\\
[T(\mathbf{x})]_2
\end{bmatrix}
=
A
\begin{bmatrix}
x_1\\
x_2
\end{bmatrix}
$$
This matrix is a function of ##\theta## alone.
 
  • #5
Thank you!
 
  • #6
Note that if you apply matrix [itex]\begin{bmatrix}a & b \\ c & d\end{bmatrix}[/itex] to the vector [itex]\begin{bmatrix}1 \\ 0 \end{bmatrix}[/itex] the result is [itex]\begin{bmatrix}a & b \\ c & d \end{bmatrix}\begin{bmatrix} 1 \\ 0 \end{bmatrix}= \begin{bmatrix}a(1)+ b(0) \\ c(1)+ d(0)\end{bmatrix}= \begin{bmatrix}a \\ c \end{bmatrix}[/itex].

That is, applying a matrix to the vector (1, 0) gives the first column of the matrix as a vector. Similarly, applying a matrix to the vector (0, 1) gives the second column of the matrix.

Now, the vector (1, 0) is 10 degrees "below" the line of reflection so its reflection is 10 degrees "above" or 20 degrees above the x-axis. What are the components of a vector of length 1 that makes an angle 20 degrees above the x-axis? The vector (0, 1) is 80 degrees "above" the line of reflection so its reflection is 80 degrees "below" the line of reflection or 70 degrees below the x- axis. What are the components of a vector of length 1 that makes an angle 70 degrees below the x-axis?
 

What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another while preserving the basic structure of the original vector space. It is a fundamental concept in linear algebra and is used to describe many real-world phenomena and relationships.

How is a linear transformation represented?

A linear transformation is represented by a square matrix. Each column of the matrix represents where a basis vector in the original vector space is mapped in the new vector space. The transformation can be applied to any vector in the original vector space by multiplying it with the matrix.

What are the properties of a linear transformation?

There are three main properties of a linear transformation: 1) preservation of addition, where the transformation of the sum of two vectors is equal to the sum of the individual transformations, 2) preservation of scalar multiplication, where the transformation of a vector multiplied by a scalar is equal to the scalar multiplied by the original transformation, and 3) preservation of the zero vector, where the transformation of the zero vector is equal to the zero vector.

What is the difference between a linear transformation and an affine transformation?

While a linear transformation maps one vector space to another, an affine transformation also includes a translation component. This means that an affine transformation can move and rotate objects in addition to scaling and shearing them. In mathematical terms, an affine transformation can be represented by a non-square matrix.

What are some applications of linear transformations?

Linear transformations have a wide range of applications in various fields, including computer graphics, physics, economics, and engineering. They are used to model and analyze systems with multiple variables, such as in optimization problems and control systems. They are also used in data compression, image processing, and machine learning algorithms.

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