Image and range of linear transformations

Get the most by viewing this topic in your current grade. Pick your course now.

?
Intros
Lessons
  1. Image and Range of Linear Transformations Overview:
  2. Matrix Transformations
    • Transforming from xx to bb
    • How transforming vector look like visually
  3. The Image of xx
    T(x)T(x): the image of xx under the transformation TT
    • Finding the image T(x)T(x) when given xx
    • Finding xx when given the image T(x)T(x)
  4. The Range of TT
    • The set of all images T(x)T(x)
    • What the range looks like visually
    • How to know if a vector is in the range of TT
?
Examples
Lessons
  1. Consider the matrix find image of a vector under t, and let's define T:R4T: \Bbb{R}^4 R3 \Bbb{R}^3 by T(x)=AXT(x)=AX. Find the images under TT of find image of a vector under t and find image of a vector under t
    1. Finding xx when given the image under TT
      Let's define T:R3T: \Bbb{R}^3 R2 \Bbb{R}^2 by T(x)=AxT(x)=Ax. Let
      find x when given the image under t

      Find the vector xx whose image under TT is bb, and find out whether xx is unique.
      1. A vector in the Range of TT
        Let's define T:R2T: \Bbb{R}^2 R3 \Bbb{R}^3 by T(x)=AxT(x)=Ax. Let
        A vector in the Range of T

        Determine if bb is in the range of the transformation TT.
        1. Geometric Interpretation of TT
          Use a graph to plot the vector vector and its image under Geometric Interpretation of T and its image under the transformation T. You are given that:
          Geometric Interpretation of T

          Explain what the transformation did to the vector.