Linear transformations and matrices

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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