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## Homework Statement

http://people.math.carleton.ca/~mezo/A8math1102-11.pdf

1b) please

1. Suppose F is a field, A ∈ Mmn(F), b ∈ Fm and v ∈ Fn is a particular solution to the equation Ax = b. Let S0 ⊆ Fn be the solution set to

the (homogeneous) equation Ax = 0, and S ∈ Fn be the solution set

to Ax = b.

(a) Prove that the set v + S0 = {v + w : w ∈ S0} is a subset of S.

(b) Prove that S is a subset of v + S0. (Hint: Suppose w ∈ S. Show that w - v ∈ S0. What does this say about w?)

## Homework Equations

## The Attempt at a Solution

How do I even know that w ε S? That's a huge supposition isn't it? Well to prove that one set is a subset of the other set, all you need to do is prove that each share a common element right?

And we know that the Set S has the following elements:

1. v+w [from the last proof]

2. v [as stated from our supposition that " v ε F^n is a particular solution to Ax=b"]

3. x [as stated from our supposition " S subset F^n be the solution set to Ax=b"

and S_0 contains the following elements in it's set:

1. 0 [trivial solution]

2. w [v+S_0) is defined as {v+w: w ε S_0}

3. x [from our supposition "Let S_0 subset F^n be the solution set to the (homogenous) equation Ax=0."

So since both have x in their elements, can't we say that S subset (v+S_0)?

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