Image and range of linear transformations

Image and range of linear transformations

Lessons

Recall the matrix equation
Ax=bAx=b

Normally, we say that the product of AA and xx gives bb. Now we are going to say that AA is a transformation matrix that transforms a vector xx into a vector bb (we call bb an image of xx).

Multiplication by A

In a sense AxAx is a function where if we plug in a vector, then it spits out another vector. If we call this function T(x)T(x), then
T(x)=AxT(x)=Ax

where TT is the transformation. Note that T(x)T(x) is an image xx since T(x)=bT(x)=b.

We say that a vector cc is in a range of the transformation TT if there exists a xx where:

T(x)=c T(x)=c
  • 1.
    Image and Range of Linear Transformations Overview:
    a)
    Matrix Transformations
    • Transforming from xx to bb
    • How transforming vector look like visually

    b)
    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)

    c)
    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


  • 2.
    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

  • 3.
    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.

  • 4.
    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.

  • 5.
    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.