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synkk
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Sorry I feel like an idiot for asking this but why is part c and b not a linear transformation? The origin would still be (0,0) and it's an expression in x and y terms so I'm confused?
thanks
Hi thank you for the response, I should of mentioned I did use to properties of linear transformations as you did in order to see that they are not linear but in the solutions they simple state that: B) is not ∵x→x^2 is not linear and c) is not ∵y→x+xy is not linear. I just don't understand how it doesn't make it linear (of course using the properties it shows it's not linear) but I'm trying to understand it in plain words of why the transformation of y --> x +xy is not linear.HallsofIvy said:They are not "linear transformations" because they are not linear. Specifically "[itex]x^2[/itex]" and "[itex]xy[/itex]" are not linear.
More precisely, a linear transformation, L, must satisfy L(x+ y)= L(x)+ L(y) and L(ax)= aLx for a any number. For b,
[tex]L\left(\begin{pmatrix}x \\ y\end{pmatrix}+ \begin{pmatrix}u \\ v\end{pmatrix}\right)= \begin{pmatrix}(x+u)^2 \\ y+ v\end{pmatrix}= \begin{pmatrix}x^2+ 2u+ u^2 \\ y+ v\end{pmatrix}[/tex]
not
[tex]L\left(\begin{pmatrix}x \\ y\end{pmatrix}\right)+ L\left(\begin{pmatrix}u \\ v\end{pmatrix}\right)= \begin{pmatrix} x^2 \\ y\end{pmatrix}+ \begin{pmatrix}u^2 \\ v\end{pmatrix}= \begin{pmatrix}x^2+ u^2\\ y+ v\end{pmatrix}[/tex]
Similarly,
[tex]L\left(a\begin{pmatrix}x \\ y \end{pmatrix}\right)= L\left(\begin{pmatrix}ax \\ ay\end{pmatrix}\right)= \begin{pmatrix}a^2x^2\\ ay\end{pmatrix}[/tex]
which is not the same as
[tex]aL\left(\begin{pmatrix}x \\ y\end{pmatrix}\right)= \begin{pmatrix}ax^2 \\ ay\end{pmatrix}[/tex]
and the same argument for (c).
HallsofIvy said:Then I have no clue what you mean by "in plain words". The "plainest" words I can use are what I said before: [itex]x^2[/itex] is not "linear" because [itex](a+ b)^2= a^2+ 2ab+ b^2\ne a^2+ b^2[/itex] and [itex]xy[/itex] is not linear because [itex](a+ b)(c+ d)= ac+ ad+ bc+ bd\ne ac+ bd[/itex].
A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the structure of the space. It is a fundamental concept in linear algebra and is used to describe many physical and mathematical phenomena.
A linear transformation can be represented using a matrix, where the columns of the matrix represent the transformed basis vectors of the input space. This matrix is known as the transformation matrix and can be used to perform the transformation on any vector in the input space.
The properties of a linear transformation include preserving the zero vector, preserving vector addition, and preserving scalar multiplication. This means that the transformation maps the zero vector to the zero vector, the sum of two vectors to the sum of their transformed counterparts, and a scalar multiple of a vector to the same multiple of its transformed counterpart.
To determine if a transformation is linear, you can check if it satisfies the properties mentioned above. Additionally, you can perform a test using two arbitrary vectors in the input space and see if the transformation preserves their linear combination. If it does, then the transformation is linear.
Linear transformations have many practical applications in fields such as engineering, physics, and computer graphics. They are used to model and describe various physical phenomena, such as rotations and translations in space, and also play a crucial role in data analysis and machine learning algorithms.