Struggling with Linear Transformation Part Two?

In summary, the conversation is about understanding the concept of diagonalizable matrices and finding the matrix S such that D = S^-1AS. The key to finding S is to solve the characteristic polynomial equation for A and use the roots to determine the columns of S. This process is an equivalence relation on matrices and can provide guidance for answering question iii.
  • #1
mslodyczka
4
0
Hi,
I'm having trouble with part two of this question. If anyone can help me out with this I would appreciate it. Thanks,
Mike
 

Attachments

  • quesito.gif
    quesito.gif
    8.4 KB · Views: 502
Physics news on Phys.org
  • #2
mslodyczka said:
Hi,
I'm having trouble with part two of this question. If anyone can help me out with this I would appreciate it. Thanks,
Mike


Some suggetions:

Understand why not all matrices are diagonalizable.

Let's assume this one is and let's call it A (i.e., A is the
representation of linear map T relative to standard basis E).

Let diagonal matrix D represent T relative to basis B.

Task: Find S such that D = S^-1AS. The columns of S are
the members of B. Done.

(Note This is an equivalence relation on matrices.
Matrices A and D are *similar*.
This should give some guidance in answering question iii.)

How to find S?
Write out the characteristic polynomial equation for A.
Solve it. The roots are key. The vectors associated with
these roots will make up the columns of S.
 
  • #3


Hi Mike,
Can you provide more details about the question or the specific part you are having trouble with? That way, I can try to assist you better. Thanks.
 

FAQ: Struggling with Linear Transformation Part Two?

What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another in a way that preserves the structure of the original space. It is a fundamental concept in linear algebra and is used in many areas of mathematics, science, and engineering.

What are the properties of a linear transformation?

The properties of a linear transformation include preserving addition and scalar multiplication, mapping the zero vector to the zero vector, and preserving the structure of vector spaces (such as collinearity and parallelism).

How do you represent a linear transformation?

A linear transformation can be represented by a matrix, which is a rectangular array of numbers. The matrix can be multiplied by a vector to obtain the transformed vector. Alternatively, a linear transformation can also be represented by a set of linear equations.

What types of transformations are considered linear transformations?

Some common types of linear transformations include rotations, reflections, dilations, and shears. Any transformation that preserves the properties of a linear transformation (as described in question 2) can be considered a linear transformation.

What are the applications of linear transformations?

Linear transformations have numerous applications in various fields, including computer graphics, signal processing, data compression, and robotics. They are also used for solving systems of linear equations and for analyzing the behavior of systems in physics and engineering.

Similar threads

I Want to understand how to express the derivative as a matrix
Replies
8
Views
2K
B What unique contributions to math did linear algebra make?
Replies
19
Views
1K
MHB Combination of Linear Transformations
Replies
2
Views
1K
I Dimension of a Linear Transformation Matrix
Replies
4
Views
2K
I Given two linear transformations L and K, show ##K = \lambda L## holds
Replies
9
Views
2K
MHB Understanding Browder's Remarks on Linear Transformations
Replies
2
Views
1K
MHB Mapping linear spaces to nonlinear ones.
Replies
1
Views
2K
I Linear Transformations: Why w1 is a Linear Combination of v
Replies
6
Views
2K
MHB Linear Transformations: Proving Rules & Demonstration
Replies
1
Views
1K
MHB 072 is Q(theta) a linear transformation from R^2 to itself.
Replies
4
Views
1K

Hot Threads

Recent Insights

Back
Top