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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