Let W be the line with parametric equations x=2 t, y=-t, z=4 t (a) Find a basis for W. (b) F

step 1 In this question, we are given a line represented by these parametric equations, and we are aske

Let W be the line with parametric equations x=2 t, y=-t, ...

Key Concepts

Orthogonal Projection

Orthogonal projection is the operation of mapping a vector onto a subspace such that the error between the vector and its projection is orthogonal to the subspace. This concept is crucial in many applications, including solving least squares problems and decomposing vectors into components parallel and perpendicular to a specified subspace.

Standard Matrix of a Projection

The standard matrix of a linear transformation, such as an orthogonal projection, is the matrix representation that describes how the transformation acts on any vector in the space. For projection onto a line spanned by a vector, the projection matrix is commonly derived using the formula (vv?)/(v?v), where v is the basis vector of the line. This matrix provides an effective way to compute the projection in a coordinate system.

Parametric Representation of a Line

A line in a vector space can be defined using parametric equations, which express each coordinate as a linear function of a parameter. This representation highlights that every point on the line is a linear combination of a fixed direction vector and a point through which the line passes. Understanding this allows one to extract the direction vector, a key element when analyzing the properties of the line as a subspace.

Basis for a Subspace

A basis of a subspace is a set of linearly independent vectors that span the subspace. For a line through the origin, any nonzero vector in the direction of the line can serve as a basis. Identifying a basis is fundamental because it provides a minimal and complete way to describe all elements in the space through linear combinations.

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