In each part, find a basis for the given subspace of R^3, and state its dimension. (a) The plane

In each part, find a basis for the given subspace of R^3,...

Key Concepts

Parametric Representation

The parametric representation of a subspace expresses every vector in the subspace as a function of free parameters. This form is particularly useful for describing the set of all solutions to a linear equation or system of equations, as it clearly identifies the degrees of freedom and facilitates the extraction of a basis by choosing specific parameter values.

Vector Subspaces

A vector subspace is a subset of a vector space that is itself a vector space under the same operations of addition and scalar multiplication. Such subspaces must contain the zero vector, be closed under vector addition, and be closed under scalar multiplication. Recognizing a subspace is fundamental in studying the structure within larger vector spaces, such as R³.

Basis of a Vector Subspace

A basis for a vector subspace is a set of vectors that are both linearly independent and span the entire subspace. This concept is crucial because the choice of a basis allows every element of the subspace to be expressed uniquely as a linear combination of the basis vectors. Finding a basis is a key step in understanding the structure of the subspace and performing coordinate transformations.

Dimension

The dimension of a vector subspace is the number of vectors in any basis for the space. It provides a measure of the 'size' or 'degrees of freedom' of the subspace, indicating how many independent directions are present. In problems involving planes and lines, the dimension helps distinguish between one-dimensional subspaces (lines) and two-dimensional subspaces (planes).

Linear Independence

Linear independence is a property of a set of vectors where no vector in the set can be expressed as a linear combination of the others. It is a critical concept in determining a valid basis, since a basis requires the vectors to be linearly independent so that each vector adds a new direction to the subspace, ensuring the uniqueness of the representation of every element in that subspace.