Determine Linear Independence of a Set of Vectors

Determine if the set of vectors $(1, -2, 0)$, $(4, 0, 8)$, $(3, -1, 5)$ is linearly independent or dependent by performing row reduction.

To determine whether a set of vectors is linearly independent or dependent, we use the method of row reduction. Linear independence is a fundamental concept in linear algebra which involves expressing vectors as linear combinations of others. Essentially, a set of vectors is considered linearly independent if no vector in the set can be written as a combination of the others. This concept is crucial as it informs us whether the vectors span a space efficiently without redundancy or overlap.

To ascertain linear independence using row reduction, we first form a matrix by treating each vector as a row. The next step involves performing elementary row operations to convert this matrix into its row echelon form or reduced row echelon form. The goal here is to simplify the matrix to a point where we can easily see if any row becomes entirely zero. A zero row suggests a linear dependency among the vectors as it indicates at least one vector can be expressed as a linear combination of others.

Understanding linear independence is vital in multiple applications within linear algebra, such as determining the basis for vector spaces, understanding dimensionality, and working with systems of linear equations. It lays the groundwork for more complex ideas such as vector spaces and spanning sets, facilitating discussions about basis and dimensionality in various mathematical contexts.