Dimension of Vector Space L(S)

In order to determine the dimension of the vector space L(S), we first need to understand the concept of the vector space and the span of a set of vectors.

Vector Space:
A vector space is a collection of vectors that satisfy certain properties. It is closed under addition and scalar multiplication, and it contains a zero vector and additive inverses. In other words, a vector space is a set of vectors that can be combined using addition and scalar multiplication.

Span of a Set of Vectors:
The span of a set of vectors is the set of all possible linear combinations of those vectors. In other words, it is the set of all vectors that can be obtained by scaling and adding the given vectors.

Given Set S:
S = {(2, 4, -3), (4, 8, -6)}

To find the dimension of the vector space L(S), we need to determine the maximum number of linearly independent vectors in the set S. Linear independence means that none of the vectors in the set can be expressed as a linear combination of the others.

Linear Independence:
To check for linear independence, we can set up a system of equations using the vectors in the set S and solve for the coefficients of the linear combination that gives the zero vector.

Let's set up the system of equations:

a(2, 4, -3) + b(4, 8, -6) = (0, 0, 0)

Simplifying the equation, we get:

(2a + 4b, 4a + 8b, -3a - 6b) = (0, 0, 0)

Now, we can write each component of the equation as a separate equation:

2a + 4b = 0
4a + 8b = 0
-3a - 6b = 0

Solving this system of equations, we find that the only solution is a = 0 and b = 0. This means that the vectors in the set S are linearly independent.

Dimension of L(S):
Since the vectors in the set S are linearly independent, the dimension of the vector space L(S) is equal to the number of vectors in the set S. In this case, there are two vectors in the set S.

Therefore, the dimension of the vector space L(S) is 2.

Conclusion:
The dimension of the vector space L(S) is 2, as the set S contains two linearly independent vectors.