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

Inverse

We can use Inverse transformation to resolve the function.

Column space

The column sapce is the span of the columns of the matrix.

And the column space represents the rank of the matrix.

When a 3-D matrix has rank 3, we can call it full rank.

Null space

For a full rank matrix only origin can be squished into origin.
But for a matrices aren’t full rank, which squish to a smaller dimension, you can have a whole bunch of vectors that land on zero.

For a 2-D matrix, if its rank is 1, it squish all the vectors on a line to the origin.

For a 3-D matrix, if its rank is 1, it squish all the vectors on a plane to the origin, if its rank is 2, it squish all the vectors on a line to the origin.

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

Determinant

Determinant means the area of the unit area after the transformation.

You can prove the formula on geometry

What does a negtive number of a determinant mean.

Calculating 3-D determinant means the volume of a parallelepiped.

The direction of a 3-D transformation satisfies right hand rule, if not the result of the determinant should be negtive.

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

3-D

Extend 2-D to 3-D, each direction has 3 elements.

Combine these transformed vector by column, we can get transformation matrix.

Rotate 90 degree on y-axis

We can also apply this transformation matrix to a vector that needs to be transformed.

3-D matrix multiplication is like doing a transformation after another in 2-D.

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