MAT211_Lecture7[1]

# MAT211_Lecture7[1] - Domain(Subset...

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1 MAT211 Image and Kernel of a Linear Transformation Domain: (Subset of)R 2 Target: (Subset of )R 4 (0,0) (1,2) (8,16) (0,0,0,0) (-1,2) (2,2,3,0) (0,5) (0,1) (10,10,15,0) (0,0,0,4) F De±nition The image of a function f : X -> Y is the subset of elements y of Y which are of the form f(x) for some x in X. The image of a function f is denoted by im(f). Example Describe the image of the linear transformation f(x,y)=(3x+6y,x+2y) De±nition Let v 1 , v 2 ,.., v m be in R n . The span of v 1 , v 2 ,.., v m is the set of all linear combinations c 1. v 1 + c 2. v 2+ ...+ c m v m We denote it by span(v 1 , v 2 ,.., v m ) Question Consider two vectors v and w in R n . Describe geometrically span(v) and span(v,w).

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2 Theorem The image of a linear transformation T(x)=Ax is the span of the columns of A. (Compare this theorem with the previous Example) Theorem: Properties of the Image Consider a linear transformation T from R m to R n The zero vector is in the image. If x and y are in the image, then x+y is in the
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MAT211_Lecture7[1] - Domain(Subset...

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