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.



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.



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.


Matrix multiplication

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


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.


Linear transformations

Transformation is like function but it focusing on moving.


  • 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 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.


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