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8. Example: A LINEAR TRANSFORMATION

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

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

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

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

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