Thread: Linear Algebra Proofs regarding subspaces and spans

1. Prove or give a counterexample to the following claim:
Claim: Let V be a vector space over the field F and suppose that W1, W2 and W3
are subspaces of V such that W1 + W3 = W2 + W3. Then W1 = W2.

2. Consider the following subspaces of the vector space R^3
over the field R of real numbers:
subspace U1, which is the plane x + y + z = 0 and subspace U2, which is the yz-plane.
a) Can R^3 be written as a sum of U1 and U2? Justify your answer.
b) Can R^3 be written as a direct sum of U1 and U2? Justify your answer.
Here x, y and z denote the usual Cartesian coordinates.


3. Let V be a vector space over the field F and suppose (v1, v2, ... , vn) is
a linearly independent set of vectors in V . Now suppose there exists w in V such
that (v1 + w, v2 + w, ... , vn + w) is a linearly dependent set of vectors in V . Prove
that w in span(v1, v2, ... , vn).

Thank you.

I would really like to see how you would at least attempt these. For example, to show that W1= W2, you must show "if vector v is in W1 then v is in W2" and "if vector v is in W2 then it is in W1". If vector v is in W1 then, for any vector, u, in W3, v+ u is in W1+ W3. Because W1+ W3= W2+ W3, v+u is in W3. Therefore, ...

For
1)
Take

Now it is true but is ?

For 3)

Consider what it means to be linearly dependent, the homogenrous eqn has atleast one non trivial (atleast one coordinate or entry is non zero)

so you have and where (index set which contains the indicies of the non zero coordinates for the homogernous eqn). or basically or Divide both sides by Why Can we Say with a gurantee that wont equal zero? Think about that one.

For
2)
a)yes
b)no

You have to justify it yourself, write out here and i can guide you.