Dimension of a Vector Space

The dimension of a vector space is the number of linearly independent vectors required to span the vector space. In other words, it is the number of basis vectors needed to represent any vector in the vector space.

Calculating the Dimension

To calculate the dimension of a vector space, we need to determine the linear independence of the given vectors. If the given vectors are linearly independent, then the dimension will be equal to the number of vectors. If the given vectors are linearly dependent, then we need to find a subset of vectors that are linearly independent and span the vector space.

Given Vectors

Let's consider the given vectors: (1, 2, 3) and (2, 4, 6).

Linear Dependence

To check for linear dependence, we can set up a system of equations and solve for the coefficients such that:

a(1, 2, 3) + b(2, 4, 6) = (0, 0, 0)

This system can be represented as:

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

Simplifying the system, we get:

a + 2b = 0
a + 2b = 0
a + 2b = 0

As we can see, the system is consistent and has infinitely many solutions. This implies that the given vectors are linearly dependent.

Reducing to Linearly Independent Vectors

Since the given vectors are linearly dependent, we need to find a subset of vectors that are linearly independent and span the vector space.

In this case, we can observe that the vector (2, 4, 6) is a scalar multiple of the vector (1, 2, 3). Therefore, the vector (1, 2, 3) alone can span the vector space.

Conclusion

Since the given vectors are linearly dependent and we can reduce them to a single vector that spans the vector space, the dimension of the vector space generated by the vectors (1, 2, 3) and (2, 4, 6) is 1.

Therefore, the correct answer is option a. 1.