What is the standard basis for R3?

The standard basis for (\mathbb{R}^3) is a set of three vectors that are linearly independent and span the entire space. In more detail, the standard basis vectors for (\mathbb{R}^3) are typically represented as ((1, 0, 0)), ((0, 1, 0)), and ((0, 0, 1)).

These vectors form a spanning set because any vector in (\mathbb{R}^3) can be expressed as a linear combination of them. For example, the vector ((x, y, z)) in (\mathbb{R}^3) can be written as:

[ x(1, 0, 0) + y(0, 1, 0) + z(0, 0, 1). ]

Additionally, the three vectors are not linearly dependent; each adds a unique direction in the three-dimensional space. Consequently, they also establish a coordinate system for (\mathbb{R}^3).

In contrast, a set of linearly dependent vectors would not form a basis for the space because