[Linear Algebra] Help with Linear Transformations part 2

In summary: The Attempt at a Solution A linear transformation ##j : V \rightarrow V## is a convenient linear map if and only if there exists a vector ##u \in V## such that##u \circ u = id_V##
  • #1
iJake
41
0

Homework Statement



Homework Statement



(a) Let ##V## be an ##\mathbb R##-vector space and ##j : V \rightarrow V## a linear transformation such that ##j \circ j = id_V##. Now, let

##S = \{v \in V : j(v) = v\}## and ##A = \{v \in V : j(v) = -v\}##

Prove that ##S## and ##A## are subspaces and that ##V = S \oplus A##.

(b) Deduce that the decomposition of the matrices in direct sum from the symmetric and skew-symmetric matrices from part (a) (finding a convenient linear transformation ##j##)

[apologies if that last part is a bit weird sounding, I'm translating from Spanish]

Homework Equations


---

The Attempt at a Solution



a) The test for ##S## and ##A## being subspaces is fairly trivial so I don't include it. Now, to determine that ##V = S \oplus A## I'm finding it a little trickier. I observe clearly that ##S \cap A = \{0\}## but how do I formalize that and lead into it proving that V is the direct sum of S and A?

b) is also confusing me. I found this and it looks remarkably similar to my problem, but I do not know how to apply it here.

Thank you Physics Forums for any help.
 
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  • #3
I apologize for taking so long to reply to this thread! I was away.

I'm still finding this one a bit tricky. At first I took ##u \in S## and ##w \in A## but these were not working for me I don't think. Now I've reached

##u \in A = j(v) + v = -v + v##
##w \in S = j(v) - v = v - v##

##v = u + w = (j(v) + v) + (j(v) - v) = (-v + v) + (v - v) = 0 + 0 = 0##

Is this the idea I'm meant to apply? I'm dubious as I don't think I really used the fact that ##j \circ j = id_V##.

I suppose once I'm sure about the methodology for this first part I'll be better equipped to deduce the relationship between symmetric and skew-symmetric matrices. What does my problem mean when it says to find the convenient linear map ##j##?

Thanks again for taking the time to assist me.
 
  • #4
check out the relatively prime decomposition theorem on page 26 of these notes;

http://alpha.math.uga.edu/%7Eroy/4050sum08.pdfit says that since your map satisfies the polynomial X^2-1 = (X-1)(X+1), then your space is a direct sum of subspaces on which your map satisfies ether X-1 or X+1.

it is not entirely trivial since the proof uses the euclidean algorithm as i recall.

or look at the equivalent decomposition lemma at the beginning of chapter 4 of these notes:

http://alpha.math.uga.edu/%7Eroy/laprimexp.pdf
 

1. What is a linear transformation?

A linear transformation is a function that maps a vector from one vector space to another in a way that preserves the vector operation of addition and scalar multiplication. In simpler terms, it is a mathematical operation that transforms one set of coordinates into another.

2. What are some common examples of linear transformations?

Some common examples of linear transformations include rotations, reflections, scaling, shearing, and projections. These transformations are commonly used in computer graphics, physics, and engineering.

3. How do you represent a linear transformation?

A linear transformation can be represented using a matrix. The transformation is applied to a vector by multiplying it with the transformation matrix. The resulting vector will have different coordinates in the new coordinate system.

4. What is the difference between a linear transformation and a nonlinear transformation?

A linear transformation follows the rules of linearity, meaning that the transformation is consistent regardless of the order in which the operations are performed. Nonlinear transformations do not follow these rules and can result in different outputs depending on the order of the operations.

5. How are linear transformations used in real life?

Linear transformations are used in various applications, including computer graphics, image processing, and data analysis. They are also used in physics to describe the movement of objects in space and in engineering to model systems and solve equations.

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