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Linear Transformation: Exploring Null Space and Range in R2

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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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