Linear Dependence of Vectors (p,q) and (r,s)

Linear dependence of two vectors (p,q) and (r,s) means that one vector can be written as a scalar multiple of the other vector. In other words, if vectors (p,q) and (r,s) are linearly dependent, there exist scalars a and b, not both zero, such that a(p,q) = b(r,s).

Explanation

When vectors (p,q) and (r,s) are linearly dependent, it implies that they lie on the same line or are collinear. Geometrically, this means that one vector is a multiple of the other, pointing in the same or opposite direction.

If vectors (p,q) and (r,s) are linearly dependent, it also means that the determinant of the matrix [p q; r s] is zero. This determinant represents the area of the parallelogram spanned by the two vectors, and if it is zero, the vectors are linearly dependent.

Consequences

1. Linearly dependent vectors do not provide independent information about a space. One vector can be expressed as a combination of the other, leading to redundancy in the information they provide.

2. Linearly dependent vectors are not a basis for the vector space. A basis requires linear independence, which is not satisfied in this case.

3. Linearly dependent vectors can complicate computations and solutions in linear algebra, as they introduce redundancies and dependencies in the system of equations they represent.

In conclusion, when vectors (p,q) and (r,s) are linearly dependent, they share a linear relationship that allows one to be expressed in terms of the other. This has implications for the information they provide, their role as a basis, and the complexity of computations involving them.