Linear transformations

• Linear transformations are also known as linear maps
• When we are working with vector spaces, linear maps are a fundamental tool to study them
  • Many natural operations such as rotation, scaling, etc. are linear maps

Definition: Linear map. A map is called a linear map (or a linear transformation) if: Where

• Examples:
  • as
    • This is called the trivial map
  • as called the identity map.
• We want to show that the identity map is actually a linear map
  • pick
  • Consider
  • So we know that this is actually a linear transformation

Basic transformations

• Say that we have some transform
  • Suppose that
  • Since the input and outputs have different dimensions, there is no identity map, so we want something similar
  • We can define a transformation
    • This is called the inclusion map.
    • This just maps all the new dimensions as 0
• Say that we have some transform ,
  • Suppose that
  • We can define a map
  • This is called a projection map
• These are the only non-degenerate maps for when the dimensions do not match
  • A degenerate transformation is one that loses information when scaling down in dimension

Remark: . We have sets of basis vectors and the image . We have some map . This is defined for basis vectors only. We want to extend the map linearly to all of . Pick some vector , . Since we know the transform for just the basis vectors, but since the transform holds the properties to be transform, we can plug in any (which is just composed of basis vectors) into the transform and know the result.

Remark: For any linear , the image of on a basis determines . That is, if is a basis and it is known that is determined.

• Scaling: say we have . Our ,
  • This would scale any set by double only on the x-axis
• Left multiplication by a matrix:
  • Pick and
    • Consider
• Say we have a map , the set
  • , this a vector
  • We can write this as
  • So you can write this as the vector of the transforms of the vectors multiplied by the coordinates of the vector
  • This is a very important example
  • This proves that all linear maps are essentially left multiplication by a matrix

Compositions

• We want to prove that the compositions of linear maps are linear
  • Suppose that we have
    • Say that we have a second map
    • is also linear
  • We can compose linear maps by multiplying them

Proof: Prove the above. Where Pick Going back to our composition: We know enough to show that As such, we know that is a composition.

Associated matrices

• Associated matrices are the matrices that when you multiply another vector, they transform them according to the transformation.

Example: Finding the matrix for the scaling linear transformation We have the map . We defined We know that

Example: Rotation, we define as . We can also write The associated matrix for this transform is:

• Suppose we have map that is linear
  • Suppose that it has basis which is the basis of
  • Say that there is a basis which is the basis of
  • Pick some element
    • When plugged into the transform,
      • We are not working with the vector itself, but its elements with respect to its basis
      • This becomes unclear when working with the standard basis
    • Since we are changing the basis to , the transform of is the coordinate with respect to the new basis
  • To summarize: , where has basis , and has basis

Definition: Linear isomorphism. A linear map is called invertible or a linear isomorphism if there exists another linear map such that Corollary: Let be a linear isomorphism. Then, Remark: . This transform is the same as the identity matrix

Midterm review

, where Let We can have . This means the coordinate representation of This means that We can also write this in a form . We can say that . Here, is our “associated matrix”, but in scalar form.

Suppose that there exists such that . What is ? We know that . Let be such that . Pick This implies that for all that this and that

Say that we have is a linear transform