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).