Define a span and a spanning set. Using the definition of a spanning set, demonstrate that(1, 1, 0), (4, 2, 1), (3, 1, 1) is not a spanning set of R3. In your answer you may use the definitions aspan and a spanning set, but not the concepts of a basis and dimension.

In linear algebra, the span of a set of vectors is the set of all possible linear combinations of the vectors. A set of vectors spans a space if their span is equal to the whole space.

A spanning set for a vector space is a set of vectors such that every vector in the space can be written as a linear combination of vectors in the set.

Now, let's consider the set of vectors {(1, 1, 0), (4, 2, 1), (3, 1, 1)}. We want to determine if this set is a spanning set for R3.

To do this, we need to see if every vector in R3 can be written as a linear combination of these vectors. A linear combination of these vectors would look like this:

a*(1, 1, 0) + b*(4, 2, 1) + c*(3, 1, 1) = (a+4b+3c, a+2b+c, b+c)

If we can choose a, b, and c such that this equation is true for any vector in R3, then the set spans R3.

However, if we look at the third component of the vector on the right side of the equation, we see that it is equal to b+c. This means that we can't choose a, b, and c to make the third component equal to any arbitrary number, because it depends on the values of b and c.

For example, if we want the third component to be 2, we can't choose b and c to be 1 and 2, because then the third component would be 3, not 2.

Therefore, the set {(1, 1, 0), (4, 2, 1), (3, 1, 1)} is not a spanning set for R3.