Calculate projections of vectors

Click For Summary
SUMMARY

The discussion focuses on calculating the projectors P1 and P2 for the vectors v1 = (cosx, sinx)ᵀ and v2 = (-sinx, cosx)ᵀ. Participants express confusion regarding the process of determining the projectors, particularly in relation to the input vector and the matrix representation of the projectors. Key questions include the roles of a1 and b1 in the context of the projectors and whether the defined projection is idempotent. The lack of clarity in the textbook is noted as a significant barrier to understanding.

PREREQUISITES
  • Understanding of vector projections in linear algebra
  • Familiarity with matrix representation of linear transformations
  • Knowledge of idempotent matrices and their properties
  • Basic trigonometric functions and their representations in vector form
NEXT STEPS
  • Study the derivation of projection matrices in linear algebra
  • Learn about the properties of idempotent matrices
  • Explore examples of vector projections in two-dimensional space
  • Review the application of trigonometric functions in vector calculations
USEFUL FOR

Students in linear algebra, mathematics educators, and anyone seeking to understand vector projections and their applications in various mathematical contexts.

ConeOfIce
Messages
13
Reaction score
0

Homework Statement


Consider the following two vectors v1= (cosx , sinx)(transpose) , v2= (-sinx , cosx)(transpose). Compute the projectors P1, P2 onto the vectors v1 and v2.


Homework Equations


(a1) (a1*,b1*) (A) <---input
(b1) ...(B)
This is a matrix
that projects on
column
(a1)
(b1)

The snowman (one bracket over the other) is supposed to be a vector.

The Attempt at a Solution



I am completely lost on this situation. The textbook is extremely unclear, and there doesn't seem to be any information on line for how you go about projectors. Am I supposed to replace the input vector with v1, and then multiply it by the matrix to see what I get, or am I supposed to solve for a1 and b1, which I presume would be the projector. Any feedback would be great

Sorry if this comes out ugly.
Thanks.
 
Physics news on Phys.org
What are a1* and b1*? Is the projection you defined idempotent?
 

Similar threads

3 Different and not parallel planes
  • · Replies 2 ·
Replies
2
Views
2K
Problem with Bernoulli calculation
  • · Replies 4 ·
Replies
4
Views
1K
Orthogonal projection onto a plane spanned by two vectors
  • · Replies 1 ·
Replies
1
Views
3K
Span and Vector Space: Understanding Vectors in Linear Algebra
  • · Replies 4 ·
Replies
4
Views
2K
How to triangulate two 3d lines
  • · Replies 1 ·
Replies
1
Views
2K
Vector addition - Vector Spaces
  • · Replies 5 ·
Replies
5
Views
2K
Show that this column matrix is not a vector
  • · Replies 2 ·
Replies
2
Views
1K
Finding perpendicular vector in a skewed coordinate system
  • · Replies 4 ·
Replies
4
Views
3K
Change of Basis Matrices for B1 and B2 in Vector Space V - Homework Solution
  • · Replies 7 ·
Replies
7
Views
4K
Using Least Squares to find Orthogonal Projection
  • · Replies 1 ·
Replies
1
Views
2K