Dimension Proof: U + W = dim(U) + dim(W) - dim(U ∩ W)

In summary, the dimension of the sum of two subspaces U and W in a vector space V is equal to the sum of their individual dimensions minus the dimension of their intersection. This can be seen visually in a Venn diagram by considering the basis vectors of each subspace. In order to prove this, one must first write a basis for the intersection of U and W, extend it to a basis for U and W separately, and then apply the Venn argument to the extended bases.
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Homework Statement


Let U, W be subspaces of a vector space V. Show that dim(U+W) = dim(U) + dim (W) - dim (U intersect W).


Homework Equations





The Attempt at a Solution


I can see this picture-wise in a venn-diagram form. In adding U and W you count the elements in their intersection twice, since both spaces contain them. Thus, you subtract the dimension of U intersect W. But, how do you show this in proof format? It doesn't seem like a verbal explanation is enough.
 
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  • #2
What you really are thinking about in your Venn diagram are basis vectors. Write a basis for U intersect W. Extend it to a basis for U and W. Now do the Venn argument.
 

Related to Dimension Proof: U + W = dim(U) + dim(W) - dim(U ∩ W)

1. What is the formula for "Dimension Proof: U + W = dim(U) + dim(W) - dim(U ∩ W)"?

The formula for "Dimension Proof: U + W = dim(U) + dim(W) - dim(U ∩ W)" is used to calculate the dimension of the sum of two vector spaces, U and W, by taking into account their individual dimensions and subtracting the dimension of their intersection.

2. How is this formula derived?

This formula is derived from the Dimension Theorem, which states that the dimension of a sum of two vector spaces is equal to the sum of their individual dimensions minus the dimension of their intersection. This can be proven using the concept of basis and linear independence.

3. Can this formula be applied to any vector spaces?

Yes, this formula can be applied to any vector spaces, as long as they are finite-dimensional and have a well-defined intersection.

4. What is the significance of this formula in linear algebra?

This formula is significant in linear algebra because it allows us to calculate the dimension of a sum of vector spaces without having to explicitly find a basis for the sum. It also provides a deeper understanding of the relationship between the dimensions of vector spaces and their intersection.

5. How is this formula used in practical applications?

This formula is commonly used in practical applications of linear algebra, such as in computer graphics, engineering, and physics. It is useful in determining the dimensions of subspaces, solving systems of linear equations, and analyzing the behavior of linear systems in real-world scenarios.

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