Proofs of subspaces in R^n (intersection, sums, etc.)

In summary, We are asked to prove statements about subspaces E and F of R^n. If the intersection of E and F is the zero vector, and we have linearly independent sets of vectors for both E and F, then the intersection of E and F is also a subspace of R^n. Additionally, the sum of E and F, where we take any vector in E and any vector in F and add them together, is also a subspace of R^n. Finally, if the intersection of E and F is the zero vector, then the dimension of the sum of E and F is equal to the sum of the dimensions of E and F. To prove these statements, we need to consider closure under addition and subtraction for the
  • #1
shellizle
4
0

Homework Statement


Let E and F be two subspaces of R^n. Prove the following statements:

(n means "intersection")
  1. If EnF = {0}, {u1, u2, ..., uk} is a linearly independent set of vectors of E and {v1, v2,...vk} is a linearly independent set of vectors
    Note: Above zero denotes the zero vector in R^n
  2. EnF = {u, such that u is in E, and u is in F} is a subspace of R^n
  3. E+F = {w=u+v, u is in E, v is in F} is a subspace of R^n
  4. If EnF={0} then dim(E+F)=dim(E)+dim(F)
 
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  • #2
If you want some help (which is probably why you're here), you need to show what you have tried to do.
 
  • #3
shellizle said:

Homework Statement


Let E and F be two subspaces of R^n. Prove the following statements:

(n means "intersection")
  1. If EnF = {0}, {u1, u2, ..., uk} is a linearly independent set of vectors of E and {v1, v2,...vk} is a linearly independent set of vectors
    Note: Above zero denotes the zero vector in R^n
  2. EnF = {u, such that u is in E, and u is in F} is a subspace of R^n
  3. E+F = {w=u+v, u is in E, v is in F} is a subspace of R^n
  4. If EnF={0} then dim(E+F)=dim(E)+dim(F)
for the intersection questions, think about closure under addition (and subtraction)
for the dimension question, think about what would happen if vectors "overlapped" in two spaces..
 

FAQ: Proofs of subspaces in R^n (intersection, sums, etc.)

What is a subspace in R^n?

A subspace in R^n is a subset of R^n that satisfies the properties of a vector space. This means that it is closed under vector addition and scalar multiplication, and contains the zero vector.

How do you prove that a subset is a subspace in R^n?

To prove that a subset is a subspace in R^n, you must show that it satisfies the three properties of a vector space: closure under vector addition, closure under scalar multiplication, and containing the zero vector. This can be done by showing that any two vectors in the subset can be added and multiplied by scalars to remain within the subset, and that the zero vector is also within the subset.

What is the intersection of two subspaces in R^n?

The intersection of two subspaces in R^n is the set of all vectors that are contained in both subspaces. This means that the intersection is a subset of both subspaces, and must also satisfy the properties of a vector space.

How do you prove the intersection of two subspaces is also a subspace in R^n?

To prove that the intersection of two subspaces is also a subspace in R^n, you must show that it satisfies the three properties of a vector space: closure under vector addition, closure under scalar multiplication, and containing the zero vector. This can be done by showing that any two vectors in the intersection can be added and multiplied by scalars to remain within the intersection, and that the zero vector is also within the intersection.

What is the sum of two subspaces in R^n?

The sum of two subspaces in R^n is the set of all vectors that can be written as the sum of a vector from one subspace and a vector from the other subspace. This means that the sum is a subset of R^n, and must also satisfy the properties of a vector space.

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