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Linear Transformation: Exploring Null Space and Range in R2

Understand the Matrix for Linear Transformations, Mapping Points, Cosets, and Domain Partitioning in R2. Learn about null spaces, ranges, and parallel lines in transformation processes.

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Linear Transformation: Exploring Null Space and Range in R2

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  2. S PHONE BOOTH

  3. S PHONE BOOTH

  4. S S S S S S S S S S PHONE BOOTH

  5. S

  6. TRANSFORMATION S

  7. Example: A LINEAR TRANSFORMATION

  8. If (x,y) is a point on y = x + 2 then Add the vector to every vector in the null space and you get a coset of the null space = the line y = x+2 = the matrix for a linear transformation from R2 into R2 The NULL SPACE of M = The RANGE of M = = points on the line y = x = points on the line y = -2x DOMAIN= R2

  9. = the matrix for a linear transformation from R2 into R2 The NULL SPACE of M = The RANGE of M = = points on the line y = x = points on the line y = -2x DOMAIN= R2

  10. = the matrix for a linear transformation from R2 into R2 The NULL SPACE of M = The RANGE of M = = points on the line y = x = points on the line y = -2x DOMAIN= R2 Every point on the line y = x + 3 is mapped to (-3,6)

  11. = the matrix for a linear transformation from R2 into R2 The NULL SPACE of M = The RANGE of M = = points on the line y = x = points on the line y = -2x DOMAIN= R2 Every point on the line y = x - 4is mapped to (4,-8)

  12. = the matrix for a linear transformation from R2 into R2 The NULL SPACE of M = The RANGE of M = = points on the line y = x = points on the line y = -2x The cosets of the null space are parallel lines that partition the domain. DOMAIN= R2 Each of these lines is mapped to a single point in the range.

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