Let W1 = L{(1, 1, 0), (−1, 1, 0)} and W2 = L{(1, 0, 2), (−1, 0, 4)}. Show that W1 +W2 = R3. Give an example of a vector v ∈ R3 such that v can be written in twodifferent ways in the form v = v1 + v2, where v1 ∈ W1, v2 ∈ W2

To show that W1 + W2 = R3, we need to show that any vector in R3 can be written as a sum of a vector in W1 and a vector in W2.

W1 is spanned by the vectors (1, 1, 0) and (-1, 1, 0), and W2 is spanned by the vectors (1, 0, 2) and (-1, 0, 4).

Let's take an arbitrary vector v = (x, y, z) in R3. We can write this vector as a sum of two vectors, one from W1 and one from W2, as follows:

v = (x, y, 0) + (0, 0, z)

The vector (x, y, 0) can be written as a linear combination of the vectors in W1. For example, if x = y, we can write (x, y, 0) = x*(1, 1, 0).

The vector (0, 0, z) can be written as a linear combination of the vectors in W2. For example, we can write (0, 0, z) = z/2*(1, 0, 2).

Therefore, any vector in R3 can be written as a sum of a vector in W1 and a vector in W2, so W1 + W2 = R3.

Now, let's find a vector that can be written in two different ways as a sum of a vector in W1 and a vector in W2.

Let's take the vector v = (2, 2, 2). We can write this vector as a sum of a vector in W1 and a vector in W2 in two different ways:

v = 2*(1, 1, 0) + 1*(1, 0, 2) = (2, 2, 0) + (1, 0, 2)

v = 1*(-1, 1, 0) + 2*(1, 0, 2) = (-1, 1, 0) + (2, 0, 4)

So, the vector v = (2, 2, 2) can be written in two different ways as a sum of a vector in W1 and a vector in W2.