Worked through Axler exercises in section 3.F, 127
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Scalar multiplication "expands" (ejude) along a "spatial" (entation) dimension.
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When looking for a linear map out of a finite dimensional space, psuedovisually pair basis vector with desired result
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Mental sense of "type matching" (direna)
Mental proof of map linearity: A shifting/sliding "transition" between equivalent formulas
Vertical stacking for tuple addition, merge downwards
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Was going to use the "define a map by how it acts on a basis" trick, felt a sense of not being confident in how that works. Felt "pressing blankness" in stomach (below xyphoid process, above navel) (patrodola).
Proving additivity mentally: Given two linear combinations of the same terms, one with constants a_{i} and the other with b_{i}, if adding them up yields a linear combination with (a_{i} + b_{i}), "slide" the bs down and over to the right entationally.
After reproving the unique existence of a map based on how it acts on basis, felt a need to talk myself through the specifics as applied to the problem at hand… No more full on patrodola but still a vague "anxiety" to confirm understanding.
Seems like uncertainty stems from "nonconstructive" assertion that the functional in question is in fact linear? Proving it helped the understanding.
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To prove linear independence, often treat each vector as symbolicentational and identify one "component" symbol (e.g. x^{2} or (1, 0)) that is only present in one vector, showing that its coefficient must be 0 if the linear combination is, and then dropping that vector and iterating
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Felt uncertain about the definition of "rank"… Is it a lack of something? Fuzziness when mentally "pulling up" the page it's defined on?
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Extended sequence of symbolically progressing/"shifting" terms one into another (valusing). Needed to work to "recall" definitions at points, work to hold "current" state active if possible. Reach end, sometimes satisfied and sometimes retrace?
Increase "the crow"/workspace! Save on paper?
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When asserting a dual basis is a basis, possible patrodola (less pressing?)… Feeling seems associated with "lookup".
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Verbalizing definitions/suppositions, a sort of entational "scan" over implications
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Almost a commerelial "conversation" walking through the proof, possibly some monitoring for a sense of validity? (didn't make any errors so unsure)