Various edits
J W
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The dimension theorem (the rank-nullity theorem) can be explained in many ways. I consider it as a consequence of the first isomorphism theorem/splitting lemma. When I teach undergrad matrix-theoretic linear algebra, I start with the equation $$Ax=b,$$ and I tell my students that the dimension theorem basically tellssays that the number of total variables equals the sum of the number of free variables and the number of "non-free" variables. They find this statement very easy. If I teach a "formal/proof-based" undergrad mathematics class, I tell my students that the dimension theorem basically tells us how much "stuff" we need to put inside the nullspaces to extend it to the given vector space.

Today I found a very good analogy: In some sense, the dimension theorem is the linear algebraic analog of the Pigeonhole Principle. Note that for any finite set $$A,$$ the function $$f: A \rightarrow A$$ is injective iff surjective iff bijective. It's a consequence of the Pigeonhole Principle. The dimension theorem gives a similar kind of conclusion for a finite-dimensional vector space $$V,$$ and any linear map $$T: V \rightarrow V.$$

Now, could you help me by providing a couple of more analogies that can be explained to an undergrad junior? Thank you so much. Please stay safe.

Various edits
J W
• 4.2k
• 2
• 17
• 41

# The dimension Theoremtheorem and pedagogy

This is a pedagogical question. I apologize if this kind of post does not go with the norm of Stackexchange.

Note that theThe dimension theorem  ( thethe rank nullity-nullity theorem) can be explained in many ways. I consider it as a consequence of the first isomorphism theorem/splitting lemma. When I teach undergrad matrix-theoretic linear algebra, I start with the equation $$Ax=b,$$ and I tell my students that dimension theorem basically tells that the number of total variables equals the sum of the number of free variables and the number of "non-free" variables. They find this statement very easy. If I teach a "formal/proof-based" undergrad mathematics class, I tell my students that the dimension theorem basically tells us how much "stuff" we need to put inside the nullspaces to extend it to the given vector space.

Today I found a very good Analogyanalogy: In some sense, the dimension theorem is the linear algebraic analog of the Pigeonhole Principle. Note that for any finite set $$A,$$ the function $$f: A \rightarrow A$$ is injective iff surjective iff bijective. It's a consequence of Pigeonhole's Principalthe Pigeonhole Principle. The dimension theorem gives a similar kind of conclusion for a finite-dimensional vector space $$V,$$ and any linear map $$T: V \rightarrow V.$$

Now, could you help me by providing a couple of more analogies that can be explained to an undergrad junior? Thank you so much. Please stay safe.

Post Migrated Here from math.stackexchange.com (revisions)
James

# The dimension Theorem and pedagogy

This is a pedagogical question. I apologize if this kind of post does not go with the norm of Stackexchange.

Note that the dimension theorem( the rank nullity theorem) can be explained in many ways. I consider it as a consequence of the first isomorphism theorem/splitting lemma. When I teach undergrad matrix-theoretic linear algebra, I start with the equation $$Ax=b,$$ and I tell my students that dimension theorem basically tells that the number of total variables equals the sum of the number of free variables and the number of "non-free" variables. They find this statement very easy. If I teach a "formal/proof-based" undergrad mathematics class, I tell my students that the dimension theorem basically tells us how much "stuff" we need to put inside the nullspaces to extend it to the given vector space.

Today I found a very good Analogy: In some sense, the dimension theorem is the linear algebraic analog of the Pigeonhole Principle. Note that for any finite set $$A,$$ the function $$f: A \rightarrow A$$ is injective iff surjective iff bijective. It's a consequence of Pigeonhole's Principal. The dimension theorem gives a similar kind of conclusion for finite-dimensional vector space $$V,$$ and any linear map $$T: V \rightarrow V.$$

Now, could you help me by providing a couple of more analogies that can be explained to an undergrad junior? Thank you so much. Please stay safe.