High School

We appreciate your visit to Find a projection matrix e which projects mathbb R 2 onto the subspace spanned by 1 1 along the subspace spanned by 1 2. This page offers clear insights and highlights the essential aspects of the topic. Our goal is to provide a helpful and engaging learning experience. Explore the content and find the answers you need!

Find a projection matrix \( e \) which projects \(\mathbb{R}^2\) onto the subspace spanned by \((1, -1)\) along the subspace spanned by \((1, 2)\).

Answer :

To project R2 onto the subspace spanned by (1, -1) along the subspace spanned by (1, 2), use the vector (1/2, -7/2) as the projection.

The subspace spanned by (1, -1) can be represented as V = {(a, -a) | a ∈ R}, and the subspace spanned by (1, 2) can be represented as W = {(b, 2b) | b ∈ R}. To find the projection vector e, we need to calculate the orthogonal projection of (1, -1) onto W.

First, we find a vector in W that is orthogonal to (1, -1). Let's call this vector w0. To find w0, we can take any vector in W and subtract its projection onto V. Choosing (1, 2) as a vector in W, we can calculate its projection onto V using the formula:

projV(1, 2) = ((1, 2) · (1, -1)) / ((1, -1) · (1, -1)) * (1, -1) = (1/2) * (1, -1).

Subtracting the projection from (1, 2), we get:

w0 = (1, 2) - (1/2) * (1, -1) = (1/2, 5/2).

Therefore, e = (1, -1) - w0 = (1, -1) - (1/2, 5/2) = (1/2, -7/2).

So, the projection vector e that projects R2 onto the subspace spanned by (1, -1) along the subspace spanned by (1, 2) is (1/2, -7/2).

Learn more about Orthogonal Projection click here :brainly.com/question/16701300

#SPJ11




Thanks for taking the time to read Find a projection matrix e which projects mathbb R 2 onto the subspace spanned by 1 1 along the subspace spanned by 1 2. We hope the insights shared have been valuable and enhanced your understanding of the topic. Don�t hesitate to browse our website for more informative and engaging content!

Rewritten by : Barada