Examples of Rank Nullity

Example

Verify conclusion of rank nullity for the transformation: $T_1:{\mathbb{R}}^3\to {\mathbb{R}}^3$ for projection to z-axis

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Text Elements

ker(T)

Image(T)

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$T_1(x,y,z)=(0,0,z)$ $dim({\mathbb{R}}^3)=3$ $dim(ker(T_1))+dim(Image(T_1))$ $=2+1=3$

Example 2

Verify conclusion of rank nullity for the transformation: $T_2:{\mathbb{R}}^2\to {\mathbb{R}}^2$ for rotation

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Text Elements

kernel

Image

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$T_2(x,y,z)=(0,0,z)$ $dim({\mathbb{R}}^2)=2$ $dim(ker(T_2))+dim(Image(T_2))$ $=0+2=2$