Dimension & Linear Maps: Does U=V?

In summary, the dimension of a space may not remain unchanged under a linear map. The rank-nullity theorem states that the dimension of the image of a linear map plus the dimension of its kernel is equal to the dimension of the original space. Therefore, the dimension of the image of a linear map can be smaller than the dimension of the original space if the map has a non-empty kernel.
  • #1
FunkyDwarf
489
0
Hey guys,

Does dimension remain unchanged under a linear map? Ie if i have a map f:U->V does dim(U) = dim(img(f))?

Cheers
-Z
 
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  • #2
Not necessarily.
 
  • #3
For a trivial example, take the zero map on a space of positive dimension.

Something that might be of interest to you is the rank-nullity theorem.
 
  • #4
Yeh i kinda figured it didnt hold. Actually i was going to use it to prove the RL theorem if it was true

cheers
Z
 
  • #5
If L is linear, L(U) cannot have dimension higher than U. It can have dimension lower than U. Of course, to do that, L must map many vectors to the 0 vector: its kernel is not empty. The "nullity-rank" theorem morphism mentioned says that the dimension of L(U) plus the dimension of the kernel of U is equal to the dimension of U.
 

1. What is the difference between a dimension and a linear map?

A dimension is the minimum number of independent variables needed to describe a mathematical space, while a linear map is a function between two vector spaces that preserves the operations of addition and scalar multiplication.

2. How can I determine if U and V are equal in terms of dimension and linear maps?

To determine if U and V are equal, you must first compare their dimensions. If the dimensions are not equal, then U and V are not equal. However, if the dimensions are equal, you must also compare the linear maps of U and V. If the linear maps are identical, then U and V are equal.

3. Can two vector spaces with different dimensions have the same linear map?

Yes, it is possible for two vector spaces with different dimensions to have the same linear map. This can occur when one vector space is a subset of the other, and the linear map simply maps all elements of the larger space to the corresponding elements in the smaller space.

4. What is the importance of U=V in the context of dimensions and linear maps?

The equality U=V is important because it signifies that two vector spaces are essentially the same, both in terms of their dimensions and their linear maps. This allows for a more efficient and effective analysis and manipulation of the spaces and their elements.

5. Are there any real-world applications of dimensions and linear maps?

Yes, dimensions and linear maps have numerous real-world applications. For example, in physics, dimensions are used to describe the physical properties of objects and linear maps are used to model physical systems. In economics, dimensions are used to represent different variables and linear maps are used to analyze relationships between these variables. In computer graphics, dimensions are used to define the size of images and linear maps are used to transform these images. These are just a few examples of the many practical applications of dimensions and linear maps.

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