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Eigenvalue and Eigenvector

Eigenvectors and Eigenvalues are those vectors don’t change direction only stretching with the multiplication of eigenvalue.

For a transformation that have eigenvectors can span the space, we can use eigenvectors as basis.

If we do so, we can apply the eigenvectors to this transformation. The result must be a diagonal matrix, and the value equal to the eigenvalues. (eigenvector only stretch on its direction, so the value only shows on the diagonal)

If we want to calculate 100-th power of this matrix, we can convert it to its system and calculate, then convert it back.

set of eigenvectors is also called eigenbasis.


Reference
https://www.3blue1brown.com/topics/linear-algebra

Change of basis

The basis in Jennifer’s grid represent a transformation as matrix below. It can convert the vector in Jennifer’s language into the vector in our language.

The inverse switches the direction.

If we apply the inversion of basis matrix to the vector in our language, we can obtain the vector in Jennifer’s language.

The function is inverse change of previous pic.

This transformation can be written in A^(-1)MA. This represent convert other language into our language, then apply a transformation to it, finally convert our language to origin language.

This transformation means a transformation on the other language.


Reference
https://www.3blue1brown.com/topics/linear-algebra

Cross product

For cross product in 2-D, it represents the area of parallelogram. And it has an order that i on the right of the j.

Actually, cross product is in 3-D, and the result of the cross product is a vector with length of the area of parallelogram, perpendicular to the parallelogram obeying right hand rule.

Cross product can be calculated as follow.

Geometry

By using duality, exist a vector p dot product [x, y, z] equals determinant.

3-D determinant means the volumn of the cube, we need to calculate component of [x, y, z] perpendicular to v and w times area of parallelogram.

This equals to the vector p with the length of area of parallelogram by using duality.


Reference
https://www.3blue1brown.com/topics/linear-algebra

Dot product

In similar directions v*w > 0

In perpendicular direction v*w = 0

In Opposing directions v*w < 0

Dot product is translate one of the vector to the transformation.

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Duality

Making a symmetric line which seperate basis vector and input vector equally.

When you try find a linear transformation to map vectors in 2-D space into number line, you will be able to match it to some vector, which is called dual vector of the transformation.


Reference
https://www.3blue1brown.com/topics/linear-algebra

Nonsquare

3 by 2 matrix, which means projecting 2 dimension to 3 dimension.

2 by 3 matrix, which means input is 2 dimension and output is 3 dimension.

1 by 2 matrix, input is a 2-D vector and output is a number. (related to dot product)


Reference
https://www.3blue1brown.com/topics/linear-algebra