Let W be the plane with equation 5 x-3 y+z=0 (a) Find a basis for W. (b) Find the standard matri

Let W be the plane with equation 5 x-3 y+z=0 (a) Find a...

Key Concepts

Subspace and Basis

A subspace of a vector space is a subset that is also a vector space under the same operations. A basis for a subspace is a minimal set of linearly independent vectors that spans the entire subspace. This concept is crucial when representing vectors within the subspace and for constructing coordinate systems relative to that subspace.

Parametrization of a Plane

Parametrizing a plane involves expressing the plane in terms of free variables, from which a basis can be derived. It is typically done by solving the plane equation for one variable in terms of the others, thus generating a set of independent direction vectors that lie within the plane.

Normal Vector

The normal vector to a plane is a vector that is perpendicular to every vector in the plane. It is directly obtained from the coefficients of the plane equation. The normal vector plays a central role in calculations involving projections and distances related to the plane.

Orthogonal Projection

Orthogonal projection refers to the process of mapping a vector onto a subspace such that the error vector (the difference between the original vector and its projection) is perpendicular to the subspace. This operation minimizes the distance between the original vector and the subspace, and it is characterized by properties such as linearity, idempotence, and symmetry.

Standard Matrix of a Linear Transformation

The standard matrix of a linear transformation provides a concrete way to express the transformation in terms of matrix multiplication. In the context of projections, the standard projection matrix transforms any vector in the space into its projection onto the specified subspace, encapsulating the projection operation in matrix form.

Transcript

00:02 In this question, we are given the scalar equation of a plane w, and we are asked to find, first of all, a basis for w, and second of all, a standard matrix for the orthogonal projection onto w.

00:27 So you may recall that we can read off the normal vector from this equation, 5, negative 3, 1, and that is because this equation is telling us that the dot product of 5, negative 3, 1 with any vector on the plane x, y, z is 0, meaning for every x, y, z on the plane, that x, y, z is perpendicular to 5, negative 3, 1.

00:57 Using this information, we can come up with two vectors, two vectors x, y, z, whose span covers all of w.

01:15 That is, every vector in w can be written as a linear combination of those vectors.

01:24 Since a plane is a two -dimensional subspace, it can be spanned by two linearly independent vectors.

01:32 In other words, a basis must have two vectors.

01:36 So in order to create one of them, let's start by giving the x -coordinate of 0 and a y -coordinate of 1, and then figure out what z would have to be.

01:51 So that gives us that z must be 3.

02:34 That's one vector that lies in the plane, but how about another one that's linearly independent to it? well, in order for two vectors not to be linearly dependent, all we have to do is make sure that they are not scalar multiples of each other.

02:58 So one way to do this is make the first coordinate a 1.

03:05 So 1 is definitely not a scalar multiple of 0, because 0 is the only scalar multiple of itself.

03:12 And we can also put a 0 in the second coordinate to make our calculations easy, and then find what the z would have to be.

03:23 So in this case, we would have 5 plus z equals 0, so z would have to be negative 5.

03:40 So those are two vectors lying in the plane that are linearly independent, meaning they form a basis for the plane.

03:50 Now let's move on to part b, where we're finding the standard matrix for the projection onto the plane.

03:59 So we know a formula for the projection, or for the standard matrix of the projection onto the column space of a matrix.

04:09 How can we use that? well, if we can express the column space of this matrix a, or rather, if we can express the plane as the column space of a matrix a, then we can use this formula, which by the way is this.

04:40 So because the column space of a matrix a is the span of its columns, and w, as we just found out, is the span of these two vectors, writing w, or sorry, writing a as a matrix whose columns are these two vectors, will equate w with the column space of a.

05:08 So now let's just compute the formula for this a...

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