The matrix of a linear transformation

The matrix of a linear transformation

Lessons

The standard basis, e1e_1 and e2e_2, are unit vectors in R2\Bbb{R}^2 such that:
Standard basis, unit vectors in r^2
If transformed vectors are the transformed vectors, then the standard matrix is
Standard matrix

Why does that work? Watch the intro video

The standard basis, e1e_1, e2e_2, and e3e_3 are unit vectors in R3\Bbb{R}^3 such that:
unit vectors of standard basic e_1, e_2, e_3

If transformed vectors e_1, e_2, e_3 are the transformed vectors, then the standard matrix is
Standard matrix e_1, e_2, e_3

To find the standard basis in R2\Bbb{R}^2 geometrically in a graph we:
1. Draw the standard basis e1e_1 and e2e_2 in the x1x2x_1 x_2 plane
2. Draw the transformed vectors using the information given
3. Identify the transformed vectors T(e1)T(e_1), and T(e2)T(e_2).
4. Combine them to get the standard matrix

Here are the many types of transformations you may see in this section:

Reflections
x1-axis reflection x2-axis reflection
x2=x1 reflection origin reflection

Vertical/Horizontal Expansions and contractions
horizontal expansion/contraction vertical expansion/contraction

Vertical/Horizontal Shears
horizontal shear vertical shear

Projections
projection onto x1 axis projection onto x2 axis

Circle Rotation
circle rotation transformation
  • Introduction
    The Matrix of a Linear Transformation Overview:
    a)
    The Standard Basis and Matrix
    T(x)=Ax,AT(x)=Ax, A: The Standard Matrix
    R2\Bbb{R}^2 standard Basis: r^2 standard basis e1 and r^2 standard basis e2
    R3\Bbb{R}^3 standard Basis: r^3 standard basis e1, e2, e3
    • Transformed standard basis
    • Finding the Standard Matrix

    b)
    Finding the Standard Matrix Geometrically in R2\Bbb{R}^2
    • Drawing the Standard basis on a Graph
    • Identifying the Transformed Standard Basis
    • Combining the Standard Basis’ into a Matrix

    c)
    Types of Geometric Linear Transformations in R2\Bbb{R}^2
    • Reflections on a Line
    • Horizontal Expansion/Contraction
    • Vertical Expansion/Contraction
    • Horizontal & Vertical Shears
    • Projections
    • Rotation Transformations


  • 1.
    Finding the Standard Matrix with Transformed Basis
    Assume that TT is a linear transformation. Find the standard matrix of TT if standard matrix of T, T(e1) standard matrix of T, T(e2) where standard matrix of T, e1= [1 0] and standard matrix of T, e2= [0 1].

  • 2.
    Assume that TT is a linear transformation. Find the standard matrix of TT if standard matrix of T, T(e1) standard matrix of T, T(e2), T(e3) where standard matrix of T, e1= [1 0 0], e2= [0 1 0] and standard matrix of T, e3= [0 0 1].

  • 3.
    Finding the Standard Matrix Geometrically
    Assume that TT is a linear transformation. Find the standard matrix of TT if T:R2T: \Bbb{R}^2 R2 \Bbb{R}^2 is a vertical shear transformation that maps e1e_1 to e1+2e2e_1+2e_2 and leaves e2e_2 unchanged.

  • 4.
    Assume that TT is a linear transformation. Find the standard matrix of TT if T:R2T: \Bbb{R}^2 R2 \Bbb{R}^2 first performs a x2=x1x_2=x_1 reflection, and then a x1x_1 reflection.

  • 5.
    Finding the Matrix Algebraically
    Show that TT is a linear transformation by finding a matrix AA that implements the mapping:
    Finding the Matrix Algebraically and show linear transformation

  • 6.
    Finding for x given the Transformation
    Let T:R2T: \Bbb{R}^2 R2 \Bbb{R}^2 be a linear transformation such that Finding for x given the Transformation. Find a xx such that Finding for x given the Transformation