Isomorphism of L_A: Orthogonal Matrix, ℝ^n -> ℝ^n

In summary: Given an invertible matrix A, the rank-nullity theorem states that there exists a matrix P such that AP is rank-nullity. So, if you can find a matrix P such that AP is rank-nullity, then P is the inverse of A.
  • #1
damabo
54
0

Homework Statement



if [itex]L_A: ℝ^n -> ℝ^n : X-> A.X [/itex] is a linear transformation, and A is an orthogonal matrix, show that L_A is an isomorphism.
also given is that [itex] (ℝ,ℝ^n,+,[.,.]) [/itex] , the standard Euclidian space which has inproduct [X,Y]= X^T.Y

Homework Equations



ortogonal matrix, so [itex]A^T=A^{-1}[/itex]
isomorphism = bijective and linear (so what is left to show is bijective)

The Attempt at a Solution



don't know where to start
 
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  • #2
damabo said:

Homework Statement



if [itex]L_A: ℝ^n -> ℝ^n : X-> A.X [/itex] is a linear transformation, and A is an orthogonal matrix, show that L_A is an isomorphism.
also given is that [itex] (ℝ,ℝ^n,+,[.,.]) [/itex] , the standard Euclidian space which has inproduct [X,Y]= X^T.Y

Homework Equations



ortogonal matrix, so [itex]A^T=A^-1[/itex]
isomorphism = bijective and linear (so what is left to show is bijective)


The Attempt at a Solution



don't know where to start

Well, what's the definition of bijective?
 
  • #3
surjective and injective ofcourse:
injective:
I must show that if [itex]X_1 != X_2[/itex] then [itex]A.X_1 ≠A.X_2.[/itex]
So choose [itex]X_1,X_2 \in ℝ^n[/itex]. because [itex] [AX_1,AX_2]=X_1^T.A^T.A.X_2 = X_1^T.X_2=[X_1,X_2] [/itex], length, distance and orthogonality will be preserved. so [itex]A.X_1 ≠A.X_2 [/itex].
surjective:
I must show that for every [itex]Y \in ℝ^n : A.X=Y [/itex]for some [itex]X \in ℝ^n [/itex]
Choose Y in ℝ^n. Is there an X in ℝ^n such that A.X=Y? I'm not sure what to do here.
 
  • #4
damabo said:
surjective and injective ofcourse:
injective:
I must show that if [itex]X_1 != X_2[/itex] then [itex]A.X_1 ≠A.X_2.[/itex]
So choose [itex]X_1,X_2 \in ℝ^n[/itex]. because [itex] [AX_1,AX_2]=X_1^T.A^T.A.X_2 = X_1^T.X_2=[X_1,X_2] [/itex], length, distance and orthogonality will be preserved. so [itex]A.X_1 ≠A.X_2 [/itex].
surjective:
I must show that for every [itex]Y \in ℝ^n : A.X=Y [/itex]for some [itex]X \in ℝ^n [/itex]
Choose Y in ℝ^n. Is there an X in ℝ^n such that A.X=Y? I'm not sure what to do here.

You've shown A is injective, so you know the kernel of A is {0}. Use the rank-nullity theorem.
 

1. What is the definition of isomorphism in mathematics?

Isomorphism is a mathematical concept that refers to a one-to-one correspondence between two mathematical objects. In other words, it is a mapping between two structures that preserves their underlying properties and relationships.

2. What does L_A represent in the isomorphism of orthogonal matrices?

L_A refers to the linear transformation that is represented by the orthogonal matrix A. This transformation maps vectors from ℝ^n to ℝ^n and preserves the dot product between vectors, as well as their lengths and angles.

3. What is the significance of an orthogonal matrix in isomorphism?

An orthogonal matrix is important in isomorphism because it represents a linear transformation that preserves distance and angles between vectors. This is crucial in many applications, such as computer graphics and data analysis, where maintaining the relative positions and orientations of objects is important.

4. How is the isomorphism of L_A related to the concept of similarity in linear algebra?

The isomorphism of L_A is closely related to the concept of similarity in linear algebra. Just like isomorphism, similarity is a one-to-one mapping between two objects that preserves their properties. In the case of linear transformations, similarity means that the two transformations have the same matrix representation, but may differ in terms of the basis used.

5. Can you give an example of the isomorphism of L_A using a specific orthogonal matrix?

One example of the isomorphism of L_A is the transformation represented by the 2x2 orthogonal matrix A = [[cosθ, -sinθ], [sinθ, cosθ]], where θ is the angle of rotation. This transformation maps any vector in ℝ^2 to a rotated version of itself, while preserving the length and angle relationships between vectors.

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