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Kernel and image

Linear maps

Japanese school year: University year 2

What you learn

Learn about the kernel, which consists of vectors mapped to zero, and the image, representing all possible outputs. These spaces are fundamental for analyzing linear transformations through the rank-nullity theorem. Prior understanding of linear maps and systems of linear equations is helpful.

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Key points

This defines the kernel (null space) of a linear map ff. It is the subspace collecting all input vectors x\boldsymbol{x} in VV that get sent to the zero vector 0\boldsymbol{0}.

Ker⁡(f)={x∈V∣f(x)=0}\operatorname{Ker}(f) = \{\boldsymbol{x} \in V \mid f(\boldsymbol{x}) = \boldsymbol{0}\}

This defines the image (range) of a linear map ff. It is the subspace of all possible output vectors reached when applying ff to every vector x\boldsymbol{x} in VV.

Im⁡(f)={f(x)∣x∈V}\operatorname{Im}(f) = \{f(\boldsymbol{x}) \mid \boldsymbol{x} \in V\}

This is the rank-nullity theorem, a fundamental relation in linear algebra. Adding the dimension of what collapses to zero (kernel) and what reaches the output (image) always equals the dimension of space VV.

dim⁡(Ker⁡(f))+dim⁡(Im⁡(f))=dim⁡(V)\dim(\operatorname{Ker}(f)) + \dim(\operatorname{Im}(f)) = \dim(V)

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