Transformations inspired by Euclid
In linear algebra, there are many "special" types of linear transformations that have names that connote concepts we have in our real world, such as reflections and projections. These concepts have been generalized to apply to all types of vectors, but the geometric description of them with Euclidean vectors gives us an idea as to why they work the way they do. This intuition can then be taken and applied to all types of vectors and vector spaces.
Translation
The first transform we will look at is translation. It transforms all vectors in a vector space by a displacement vector. More precisely:
In the following graph, the vector |x⟩ is translated to the right by |d⟩ to form T(|x⟩):
Figure 5.10 – A graphical depiction of translation
What's interesting about this type of translation is that it turns out to be non-linear! I will quickly show you.
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