Linear Algebra - Bilinear Forms and Change of Basis

Then, you can create the matrices Q and P by putting the coordinates of the basis vectors of one basis in the columns of the matrix. In summary, to find the matrix of f relative to Alpha' and Beta', you need to express the vectors of one basis in terms of the other basis and use them to create the matrices Q and P.
  • #1
TorcidaS
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Homework Statement


Find the matrix of f relative to Alpha' and Beta'.

Alpha' = [(1,0,0), (1,1,0), (2,-1,1)]
Beta' = [(-13,9), (10,-7)]


The question originally reads that f is a bilinear form.

I've found a (correct according to answer key) matrix A that is

3 -4
4 -5
-1 2

from a given basis of
Alpha = [(1,0,0), (1,1,0), (1,1,1)] and
Beta = [(1,-1), (2,-1)]

Homework Equations



If Q is the matrix of transition from Alpha to basis Alpha' of U and P is the matrix of transition from Beta to basis Beta' of V, then QTAP = matrix of f relative to Alpha' and Beta'.
T meaning transpose.

The Attempt at a Solution


I've gotten oh so far with this question, but I'm stuck here in the final part. I'm confused how to attain said matrices Q and P from the relevant equations. If I'm not terrible mistaken, I think I have to do something involving inverses but I'm at a loss here.

Any help will of course be greatly appreciated.
 
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  • #2
You want to express the vectors of one basis in terms of the other basis.
 

1. What is a bilinear form in linear algebra?

A bilinear form is a function that takes two vectors as inputs and produces a scalar as its output. It is a type of multilinear form, which means it is linear in each of its two arguments. Bilinear forms are commonly used to describe relationships between vectors and can be represented by matrices.

2. How is a bilinear form different from a linear transformation?

A linear transformation is a function that takes a vector as an input and produces another vector as its output. In contrast, a bilinear form takes two vectors as inputs and produces a scalar as its output. While both are linear in their respective arguments, a linear transformation is a function of one variable, whereas a bilinear form is a function of two variables.

3. What is the significance of the change of basis in linear algebra?

The change of basis is a fundamental concept in linear algebra that allows us to express vectors and linear transformations in different coordinate systems. It provides a way to represent the same vector or transformation using different basis vectors, which can be useful in solving problems that involve different coordinate systems.

4. How is the change of basis related to bilinear forms?

When we change the basis of a bilinear form, we are essentially changing the coordinate system in which the form is expressed. This allows us to manipulate and analyze the form in different ways and can help us gain a better understanding of its properties. In some cases, changing the basis can also make it easier to solve problems involving bilinear forms.

5. Can bilinear forms be used in real-world applications?

Yes, bilinear forms have many real-world applications, particularly in physics and engineering. They are used to model physical systems and describe relationships between variables. For example, in mechanics, a quadratic form (a type of bilinear form) can represent the kinetic energy of a system, and the eigenvectors of this form can correspond to the principal axes of rotation of the system.

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