# Linear Algebra - adding subspaces

• moonbounce7
In summary, the sum of two subspaces U and U is simply U itself. This can be justified by the definition of addition among subspaces and the fact that U is a subspace.
moonbounce7
1. Suppose that U is a subspace of V. What is U+U?

2. Homework Equations :
There's a theorem that states: Suppose that A and B are subspaces of V. Then V is the direct sum of A and B (written as A [plus with a circle around it] B) if and only if: 1) V=A+B (meaning, the two subspaces are technically able to be added), and 2) The intersection of A and B = {0}.

3. I don't understand what the addition symbol really means in this case... I know that addition of two subspaces is NOT the same as the union of two subspaces. The union of two subspaces of V is a subspace of V if and only if one of the subspaces is contained in the other. ...Help!

The sum of two subspaces A and B is the set of all sums a + b such that a is in A and b is in B. The direct sum of two subspaces A and B as subspaces of V is the set of all vectors in V that can be written uniquely as ka + jb for k,j in R (if R is the field), a in A and b in B. This is the one that is usually denoted as a circle around the plus sign.

Thanks for trying to help, I think I've figured it out though:

The definition of addition among subspaces follows this definition:

U+U = {a+b, such that aЄU and bЄU}.

Since U is a subspace, addition is closed in U, so adding two elements in U would simply produce another element in U.

Therefore, U+U = U.

A justification would be a formal proof such that U ⊆ U+U and U+U ⊆ U.

## 1. What is a subspace in linear algebra?

A subspace in linear algebra is a subset of a vector space that satisfies the same properties as the vector space itself. This means that a subspace must contain the zero vector, be closed under vector addition and scalar multiplication, and contain all linear combinations of its vectors.

## 2. How do you add two subspaces in linear algebra?

To add two subspaces in linear algebra, you can use the intersection method or the direct sum method. In the intersection method, you find the common vectors between the two subspaces and combine them to create a new subspace. In the direct sum method, you combine the two subspaces without overlapping any vectors.

## 3. Can you add more than two subspaces in linear algebra?

Yes, you can add more than two subspaces in linear algebra. You can use the intersection method or the direct sum method to combine multiple subspaces. However, as the number of subspaces increases, the calculations become more complex.

## 4. What is the difference between adding subspaces and multiplying subspaces in linear algebra?

Adding subspaces is the process of combining two or more subspaces to create a new subspace. On the other hand, multiplying subspaces involves finding the space spanned by the products of all possible combinations of vectors from the two subspaces. While adding subspaces results in a subspace, multiplying subspaces may result in a larger or smaller space than the original subspaces.

## 5. How do you know if two subspaces can be added in linear algebra?

In order for two subspaces to be added in linear algebra, they must have the same dimension. This means that the number of vectors in each subspace must be equal, and the vectors must be linearly independent. If the two subspaces have different dimensions, they cannot be added.

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