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

[shellizle]

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)

 

[Mark44]

If you want some help (which is probably why you're here), you need to show what you have tried to do.

 

[emyt]

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..