What is the Significance of (I-P)(x) in Linear Algebra?

  • Thread starter transgalactic
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In summary, you are projecting x onto R. If some vector, y, is already pointing in the R direction, P(y)= y. If you project again, since xR is already along R, projecting again just gives you the same thing again. As for P(xN)= 0, that is only true if N is orthogonal to R. Your picture does not make it look like that but you say "ker P= N". If xN is in ker P, then by definition of kernel, P(x)= 0. Are you sure you aren't told somewhere that R and N are orthogonal? Finally, (I- P)x= Ix- P(x)= x
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  • #2
Your two questions are "Why are P(xR)= xR and P(xN)= 0?" and "what is the meaning of (I- P)(x)?".

You are projecting x onto R. That is, xR= P(x), the component of x lying along R. If some vector, y, is already pointing in the R direction, P(y)= y. If you project again, since xR is already along R, projecting again just gives you the same thing again.

As for P(xN)= 0, that is only true if N is orthogonal to R. Your picture does not make it look like that but you say "ker P= N". If xN is in ker P, then by definition of kernel, P(x)= 0. Are you sure you aren't told somewhere that R and N are orthogonal?

Finally, (I- P)x= Ix- P(x)= x-P(x). It is x, with P(x) subtracted. If you drop a perpendicular from the tip of x to R, the intersection is at the tip of R- you have a right triangle with hypotenuse x, "near side" P(x), and opposite side x- P(x). If you "follow" the path from the origin of x to its tip, then down that perpendicular (call it "y"), you get to the tip of P(x): x+ y= P(x) so y= P(x)- x= -(I- P)x: -(I- P)x and so (I-P)x is always perpendicular to R.
 
  • #3
can you please make a drawing to explain this p(Xr) p(Xn)
because i can't emagine all this components
and you said aslo that's my drawing is not correct

regarding the second part i didnt understand
the theoretical part of it
why are they doing
(I- P)(x)
what is the theoretical meenig of that
 
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  • #4
If you did not understand what I said before, then I recommend you talk to your teacher. Apparently you do not understand what a "projection" is.
 
  • #5
correct me if i got it wrong p is taking the parts of the given vector
which is located on R

then the the axes are supposed to be perpandicular

i still can't understand the meening of the sign:
why are they doing
(I- P)(x)
what is the theoretical meenig of that
??
 
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FAQ: What is the Significance of (I-P)(x) in Linear Algebra?

What is linear algebra?

Linear algebra is a branch of mathematics that deals with vector spaces and linear transformations. It involves the study of systems of linear equations, matrices, and vector spaces, and their properties and operations.

Why is linear algebra important?

Linear algebra has a wide range of applications in various fields such as physics, engineering, computer science, and economics. It provides a powerful and efficient framework for solving complex problems involving large sets of data and equations.

What are the basic concepts of linear algebra?

The basic concepts of linear algebra include vectors, matrices, linear transformations, determinants, eigenvalues and eigenvectors, and systems of linear equations. These concepts form the foundation for understanding more complex topics in linear algebra.

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What are some real-life applications of linear algebra?

Linear algebra has numerous real-life applications, including image and signal processing, cryptography, network analysis, and optimization problems. It is also used in the fields of computer graphics, robotics, and quantum mechanics.

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