Linear Algebra Transition Matrix Proof

In summary, The theorem states that if B, C, and D are ordered bases for a nontrivial finite dimensional vector space V, and P is the transition matrix from B to C, and Q is the transition matrix from C to D, then QP is the transition matrix from B to D. This means that for every vector v in V, P[v]B=[v]C and Q[v]C=[v]D. The proof for this theorem involves showing that QP is equivalent to the transition matrix from B to D, which can be represented as the ith column of [bi]D. However, further steps may be needed to fully understand and prove this theorem.
  • #1
lkyabber
2
0

Homework Statement



Prove the following theorem:
Suppose that B, C, and D are ordered bases for a nontrivial finite dimensional vector space V. let P be the transition matrix from B to C, and let Q be the transition matrix from C to D. Then QP is the transition matrix from B to D.

Homework Equations



For every v contained in V: P[v]B=[v]c
For every v contained in V: Q[v]c=[v]D

3. The Attempt at a Solution

Not sure how to go about doing this proof..don't even know where to start. Any help is greatly appreciated.
 
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  • #2
Write out what you think QP could look like
 
  • #3
Would it be the ith column of [bi]D?
But I'm still not sure what that looks like
 

FAQ: Linear Algebra Transition Matrix Proof

1. What is a transition matrix in linear algebra?

A transition matrix in linear algebra is a square matrix that represents a linear transformation between two vector spaces. It maps the coordinates of a vector from one basis to another.

2. How is a transition matrix calculated?

A transition matrix can be calculated by applying a linear transformation to a standard basis for the vector space. The resulting coordinates of the transformed basis vectors make up the columns of the transition matrix.

3. What is the purpose of proving the properties of a transition matrix?

Proving the properties of a transition matrix helps to understand how it affects the vector space and how it behaves under different operations. This is essential in solving problems and making accurate predictions in various fields such as physics, engineering, and computer graphics.

4. What are some common properties of a transition matrix?

Some common properties of a transition matrix include being invertible, having a determinant of non-zero value, and preserving vector addition and scalar multiplication. It also has a unique solution for a linear system of equations.

5. How do you prove the properties of a transition matrix?

The properties of a transition matrix can be proven using various techniques such as using the definition of a linear transformation, using the properties of matrix operations, and using mathematical induction. Additionally, using concrete examples and counterexamples can also help in proving the properties.

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