let V=R^3 and S={e1,e2,e3}.prove that S is a basis of V

To prove that S is a basis of V, we need to show two things:

1. S spans V: This means that every vector in V can be written as a linear combination of the vectors in S.

2. S is linearly independent: This means that no vector in S can be written as a linear combination of the other vectors in S.

Let's start with the first condition:

To show that S spans V, we need to show that any vector in V can be written as a linear combination of the vectors in S (e1, e2, and e3).

In this case, V = R^3, which represents the set of all 3-dimensional vectors. Each vector in V can be written as (x, y, z), where x, y, and z are real numbers.

Now, let's consider an arbitrary vector in V: (a, b, c).

To prove that S spans V, we need to find scalars m1, m2, and m3 such that:

m1 * e1 + m2 * e2 + m3 * e3 = (a, b, c)

Since e1 = (1, 0, 0), e2 = (0, 1, 0), and e3 = (0, 0, 1), the equation becomes:

m1 * (1, 0, 0) + m2 * (0, 1, 0) + m3 * (0, 0, 1) = (a, b, c)

Simplifying further, we get:

(m1, 0, 0) + (0, m2, 0) + (0, 0, m3) = (a, b, c)

This is equivalent to the following system of equations:

m1 = a
m2 = b
m3 = c

Since this system of equations has a unique solution for any (a, b, c), we can conclude that any vector in V can be written as a linear combination of the vectors in S.

Now, let's move on to the second condition:

To show that S is linearly independent, we need to show that no vector in S can be expressed as a linear combination of the other vectors in S.

In this case, S = {e1, e2, e3}.

To prove linear independence, we need to show that the equation:

m1 * e1 + m2 * e2 + m3 * e3 = (0, 0, 0)

only has the trivial solution m1 = m2 = m3 = 0.

Using the values of e1, e2, and e3, the equation becomes:

m1 * (1, 0, 0) + m2 * (0, 1, 0) + m3 * (0, 0, 1) = (0, 0, 0)

Which simplifies to:

(m1, m2, m3) = (0, 0, 0)

Since the only solution to this equation is the trivial solution, we can conclude that S is linearly independent.

Hence, we have shown that S is both spanning and linearly independent, which means that S is a basis of V (R^3).