Today’s glossary provoked by Michael Spivak.

tensors spivak diff geom vol 1 ch 4

Below I’ve drawn two vector spaces connected by a linear homomorphism ƒ, plus a linear functional λ going to ℝ. After seeing these pictures I hope it’s easier to understand how the pullback ƒ* works.

Here’s one of the main pictures for flavour:

dual space linear functional

Also, you can probably just skim the pictures and get the point (especially the Number Field one and the final one). That’s the fastest way to read this post.

Start with an abstract 𝓥ector space.

abstract vector space

I’ll do some violence because I’ll need coordinates in a minute.

invent linear functional

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Misty Mountains
An old painting I found while doing some spring cleaning


Misty Mountains

An old painting I found while doing some spring cleaning


For what is the theory of determinants? It is an algebra upon algebra; a calculus which enables us to combine and foretell the results of algebraical operations, in the same way as algebra itself enables us to dispense with the performance of the special operations of arithmetic. All analysis must ultimately clothe itself under this form.

I have in previous papers defined a ‘Matrix' as a rectangular array of terms, out of which different systems of determinants may be engendered, as from the womb of a common parent; these cognate determinants being by no means isolated in their relations to one another , but subject to certain simple laws of mutual dependence and simultaneous deperition.

James Joseph Sylvester, 1851


(a relevant fact is Cramer’s rule: knowing only the determinants of submatrices you can find the eigenvectors, which are the stable fixed-points under the matrix operation)

from the same source, quoting Sylvester’s Apotheosis of Algebraical Quantity (1884):

A matrix … regarded apart from the determinant … becomes an empty schema of operation, … only for a moment looses the attribute of quantity to emerge again as quantity, … of a higher and unthought-of kind, … in a glorified shape-as an organism composed of discrete parts, but having an essential and undivisible unity as a whole of its own .… The conception of multiple quantity thus rises on the field of vision.

The romantic view expressed there doesn’t sound very different, to me, of Deleuze on calculus


adults, unlike children, rarely cry in public. They wait until they’re in the privacy of their homes—when they are alone or, at most, in the company of one other adult. On the face of it, the “crying-as-communication” hypothesis does not fully hold up, and it certainly doesn’t explain why we cry when we’re alone, or in an airplane surrounded by strangers we have no connection to…

In the same 2000 study, Vingerhoet’s team also discovered that, in adults, crying is most likely to follow a few specific antecedents. When asked to choose from a wide range of reasons for recent spells of crying, participants in the study chose “separation” or “rejection” far more often than other options, which included things like “pain and injury” and “criticism.”

about a paper by Vingerhoets, Cornelius, Heft, Beck

Towards a Model of Adult Crying


yes trees opening up like a scream yes the wolves moving their bodies before their own shadows yes claws falling in love with the air yes smoke always moving in the direction opposite of our bodies yes down yes a tearing in the ground yes a dream of a throne of birds yes throwing our bones upward saying yes that is my shape yes that is my dream yes stepping out of the fire with a mouth full of snow yes angels building houses in the screams yes a goodness yes oh my god yes my spine yes i think my spine is more of a spine and less of a shiver yes less of a fire more of a flowering yes less of a knife more of a knife yes raising a body like a sharp deliberate thing yes towards the sky yes delighting in whatever falls from it like a rain yes like a fire yes like a frost yes whatever falls it will not be me yes it will not be the trees yes it will not be the fur yes it will not be you yes i am more of an opening more of a deliberate thing
aheartlikea-socket, via tiny ghost hands 

(Source: Flickr / joshuamellin)

If your vector space is a shopping cart full of groceries, then the checkout clerk is a linear operator on that space.

Moises Mahiques


[F]or leaders, it’s important that people know you are consistent and fair in how you think about making decisions and that there’s an element of predictability. If a leader is consistent, people on their teams experience tremendous freedom, because then they know that within certain parameters, they can do whatever they want. If your manager is all over the place, you’re never going to know what you can do, and you’re going to experience it as very restrictive.

[Employees should be saying that] the manager treats me with respect, the manager gives me clear goals, the manager shares information, the manager treats the entire team fairly.