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Example: A LINEAR TRANSFORMATION
2211
M
2211
M = the matrix for a linear transformation from R2 into R2
The NULL SPACE of M =
11 The RANGE of M =
21
= points on the line y = x = points on the line y = -2x
DOMAIN= R2
Add the vector
31
to every vector in the null space and you geta coset of the null space = the line y = x+2
If (x,y) is a point on y = x + 2 then
42
yx
M
2211
M = the matrix for a linear transformation from R2 into R2
The NULL SPACE of M =
11 The RANGE of M =
21
= points on the line y = x = points on the line y = -2x
DOMAIN= R2
2211
M = the matrix for a linear transformation from R2 into R2
The NULL SPACE of M =
11 The RANGE of M =
21
= 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)
2211
M = the matrix for a linear transformation from R2 into R2
The NULL SPACE of M =
11 The RANGE of M =
21
= points on the line y = x = points on the line y = -2x
DOMAIN= R2
Every point on the line y = x - 4 is mapped to (4,-8)
Each of these lines is mapped toa single point in the range.
The cosets of the null spaceare parallel lines that partition the domain.
2211
M = the matrix for a linear transformation from R2 into R2
The NULL SPACE of M =
11 The RANGE of M =
21
= points on the line y = x = points on the line y = -2x
DOMAIN= R2
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