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# 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

# Matrix multiplication

The geometry of matrix multiplication is applying a transformation and then applying another transformation.

# Composition

The first transformation

The second transformation

Treat the first transformation in column, the first column is where the x vector lands, the second vector is where the y vector lands. And then apply the second transformation on them seperately.

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

# Linear transformations

Transformation is like function but it focusing on moving.

Linear

• lines remain lines
• origin remains fixed

Grid lines remain paralled and evenly space

# How to think about transformation

Before transformation we need to record where the basis vectors land. v vector is -1 * i + 2 * j

After transformation we also need to record where the transformed basis vectors land. v vector is -1 * i(transformed) + 2 * j(transformed)

# Matrices

Matrices can be thought of a set of transformed i and transformed j in columes.

To calculate where the transformed vector lands, we just need to multiply x by first column plus multiply y by second column.

Matrices is a way of package information needed to desctibe a linear transformation.

This also shows why a matrix multiply a vector in this way, the first element is each elements in the first row multiply by corresponding each elements in the first column of the vector.

# Specific

The transformation of 90 degree rotation counterclockwise.

The transformation of Shear

If the columns are linearly dependent, all the transformed vectors will be squeezed in to a line.

• 2-D space is squeezed into a line
• the span of these two linearly dependent is 1-D

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