Determining Dimension of Span in R2 with Row Reduction

Given a set of four vectors in $\mathbb{R}^2$, put them as columns of a matrix, row reduce, and identify the number of pivots to determine the dimension of the span.

In this problem, we are tasked with analyzing vectors in the two-dimensional space known as R2. This type of problem involves some fundamental concepts in linear algebra related to span, linear independence, and matrix operations.

By organizing the vectors as columns in a matrix, you essentially format them for row reduction, a systematic process that simplifies matrices to reveal their pivotal elements. These pivotal elements, or leading ones in the row echelon form, are instrumental in determining the rank of the matrix.

The rank, especially in the context of matrices whose columns are vectors from R2, tells us the dimension of the subspace spanned by these vectors. If you achieve a row-reduced form that maximizes non-zero rows, you can be confident that these correspond to independent vectors, and hence they span a subspace whose dimension equals the number of pivot columns.

Determining the number of pivots not only provides the dimension of the span but also helps understand whether vectors are linearly independent or dependent. In practical applications, this concept aids in compressing information by eliminating redundancy without losing the essence of data relationships.

This exercise hones your ability to carry out matrix row reduction effectively and enhances your reasoning about vector spaces and their dimensions—a skill crucial across many applications in fields such as computer graphics, machine learning, and physics.

By grasping these abstract concepts, you prepare to tackle increasingly complex scenarios where understanding the underlying structure of data is key.