10. Find a kernel and describe the lost information
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Consider the real linear map from three coordinates to two given by
\[T=\begin{pmatrix}1&1&0\\0&1&1\end{pmatrix},\qquad
T(x_1,x_2,x_3)^{\mathsf T}=(x_1+x_2,x_2+x_3)^{\mathsf T}.
\]
Find the erased directions
Set both outputs to zero. The equations are \(x_1+x_2=0\) and \(x_2+x_3=0\). Writing \(x_2=t\) gives
\[x=(-t,t,-t)^{\mathsf T}=t(-1,1,-1)^{\mathsf T}.
\]
Thus the kernel has basis \((-1,1,-1)^{\mathsf T}\) and dimension one. The map is not injective.
Find the image
The first and third columns are \((1,0)^{\mathsf T}\) and \((0,1)^{\mathsf T}\). They span the entire target, so \(T\) is surjective and its rank is two. The dimensions satisfy
\[\dim\mathbb R^3=3=2+1=\mathrm{rank}T+\mathrm{nullity}T.
\]
Describe every input with the same output
For output \((5,7)^{\mathsf T}\), choose \(x_2=t\). Then
\[x=(5-t,t,7-t)^{\mathsf T}
=(5,0,7)^{\mathsf T}+t(-1,1,-1)^{\mathsf T}.
\]
For instance, \((5,0,7)^{\mathsf T}\) and \((3,2,5)^{\mathsf T}\) give the same output. The entire fiber is one translate of the kernel. In the quotient space \(\mathbb R^3/\ker T\), these inputs represent the same coset.
This describes precisely what this linear observation loses. An arbitrary nonlinear observation need not have fibers that are translates of a subspace.
Try it
Describe all inputs sent to \((1,-1)^{\mathsf T}\).
Answer: \((1-t,t,-1-t)^{\mathsf T}\) for arbitrary real \(t\). Setting \(t=1\) gives \((0,1,-2)^{\mathsf T}\), whose output is \((1,-1)^{\mathsf T}\).
Papers, proofs, Lean, and code
Terms in this example: Kernel · Image · Rank · Nullity · Rank–nullity identity · Injective map · Surjective map · Quotient space · Subspace
These finite linear calculations give background for typed realizations and information loss. Read each work for its own claims.
Paper I: Typed Operator Systems, Seed Rank, and Exact Packet Coordinates
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Information Realization
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Related library guide and code examples. The guide covers related constructions and their software interfaces.
Python arithmetic check for this example: kernels(). This checks the displayed rational, modular or finite-set calculations exactly.
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