Change of basis [ x ] B + [ y ] B = [ x+y ] B[

Thread starter Maxwhale
Start date Nov 9, 2008
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Basis Change Change of basis

In summary, a change of basis is a mathematical process used to represent vectors in different coordinate systems. It involves finding the coordinates of the vector(s) in the new basis and expressing them in terms of the old basis. This is done by finding a transformation matrix and using it to convert the coordinates. It is important in understanding and analyzing vectors and transformations in different coordinate systems and has applications in areas such as physics, engineering, and computer graphics. However, it has limitations such as only working for linear transformations and requiring a good understanding of linear algebra.

Nov 9, 2008

Maxwhale

change of basis [x]_B + [y]_B = [x+y]_B[_{Homework Statement

Let B = {v1,...vn} be a basis for a vector space V, and let x = a1v1 + ... anvn and y = b1v1 + ...+ bnvn be arbitrary vectors in V.

Find [x]_B, [y]_B and [x+y]_B

Homework Equations

[x]_B + [y]_B = [x+y]_B

The Attempt at a Solution

No clue}

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Nov 9, 2008

Hurkyl

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Definitions are usually a good place to start.

Related to Change of basis [ x ] B + [ y ] B = [ x+y ] B[

What is a change of basis?

A change of basis is a mathematical process used to represent a vector or a set of vectors in a different coordinate system or basis. It involves finding the coordinates of the vector(s) in the new basis and expressing them in terms of the old basis.

How is a change of basis performed?

To perform a change of basis, you first need to have two bases: the original basis (usually denoted by B) and the new basis (usually denoted by B'). Then, you need to find a matrix P that represents the transformation from the old basis to the new basis. Finally, you can use this matrix P to convert the coordinates of the vector(s) from the old basis to the new basis using the formula [x']B' = P[x]B, where [x]B represents the coordinates of the vector x in the old basis.

Why is a change of basis important?

A change of basis is important because it allows us to understand and analyze vectors and transformations in different coordinate systems. It also helps to simplify complex calculations and make them more intuitive. In many areas of science, such as physics, engineering, and computer graphics, a change of basis is a fundamental concept that is used extensively.

What are some applications of a change of basis?

Some common applications of a change of basis include solving systems of linear equations, performing rotations and translations in 3D space, and representing geometric objects in computer graphics. It is also used in quantum mechanics to describe the state of a particle in different reference frames.

Are there any limitations to a change of basis?

While a change of basis is a powerful tool, it does have some limitations. It only works for linear transformations, and it can be difficult to find the appropriate transformation matrix for certain non-standard bases. It also requires a good understanding of linear algebra to perform accurately.