# Lin. Alg. Projections conceptual question

• Jarvis323
In summary, the inner product of x with Py is equal to the inner product of Px with y because P is a projection matrix onto the line through a, which means it projects any vector onto the line and the perpendicular component is removed. Therefore, the inner product of x with Py is the same as the inner product of x with the projection of y onto the line, which is Px. This is also true for the inner product of Px with y.
Jarvis323

## Homework Statement

16. Suppose P is the projection matrix onto the line through a.
(a) Why is the inner product of x with Py equal to the inner product of Px with y?
(b) Are the two angles the same? Find their cosines if a = (1;1;¡1), x = (2;0;1),
y = (2;1;2).
(c) Why is the inner product of Px with Py again the same? What is the angle
between those two?

## Homework Equations

Px = (a^t x / a^t a)a
Py = (a^t y / a^t a)a

## The Attempt at a Solution

I tried to multiply out the general vectors of size n and then do their dot product, but it got way to complicated.

I'm not sure which angles they want me to find the angles between. Obviously the angle between Px and Py is 0. But the angles between x and y could be arbitrary ( not talking about the specific ones in part b ), and I can't see which angles between which vectors should make it obvious that part a is true.

tAllan said:

## Homework Statement

16. Suppose P is the projection matrix onto the line through a.
(a) Why is the inner product of x with Py equal to the inner product of Px with y?

Is P given to be an orthogonal projection? If so try writing ##x = u_1 + v_1## where ##u_1## is on the line and ##v_1## is perpendicular to it. Similarly for ##y##. Then try working out those inner products, without getting down to the component level.

1 person

## What is a projection in linear algebra?

A projection in linear algebra is a transformation that maps a vector onto a subspace, resulting in a vector that lies within that subspace. This can be thought of as a shadow cast by the vector onto the subspace.

## What is the difference between orthogonal and oblique projections?

Orthogonal projections preserve the lengths and angles of the vectors, while oblique projections do not necessarily preserve these properties. In other words, an orthogonal projection is a specific type of oblique projection where the angle between the projected vector and the subspace is 90 degrees.

## Can a vector be projected onto a subspace that is not orthogonal to it?

Yes, a vector can be projected onto any subspace, regardless of its orientation to the vector. However, the resulting projection will not be an orthogonal projection unless the subspace is also orthogonal to the vector.

## What is the geometrical interpretation of a projection matrix?

A projection matrix is a square matrix that represents a projection transformation in linear algebra. Geometrically, it can be thought of as a transformation that projects points onto a subspace, similar to a shadow cast by the points onto the subspace.

## How are projections used in real-world applications?

Projections have various applications in fields such as computer graphics, statistics, and data analysis. For example, in computer graphics, projections are used to create realistic shadows and lighting effects. In statistics, projections are used to simplify data and reduce the dimensionality of a dataset. In data analysis, projections can be used to visualize high-dimensional data in a lower-dimensional space.

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