3
votes
2answers
107 views

Why is this a good picture of a covector?

I'm reading a book about applied differential geometry and the author says: "suppose $V$ is a finite dimensional vector space. For a given covector $\omega \in V^\ast$, the set $\hat{\omega}$, of ...
1
vote
0answers
22 views

Motivation behind Definition of Projection [Poole P27]

In the long paragraph above equation $(2)$, http://mathinsight.org/dot_product avers: This leads to the definition that the dot product $\mathbf{a⋅b}$, divided by $∥\mathbf{b}∥$ (= magntitude of ...
6
votes
0answers
96 views

Infer distance from a point to a line, from the distance from a point to a plane [Stewart P793 12.4.44]

I'm able to prove $44$, but how would one deduce $43$ from it without further industry, forthwith? $43$ seems like a reduced, 2D version of $44$? I'm not enquiring about individual proofs. $44.$ ...
1
vote
2answers
57 views

Visualise all vectors perpendicular to one vector and two vectors in R^3 [Strang P19 1.2.6]

I'm only asking about visual/geometric solutions herein. (b) The vectors perpendicular to any vector in $\mathbb{R^3}$ lie on what?. (c) The vectors perpendicular to any two vectors in $\mathbb{R^3}$ ...
5
votes
1answer
68 views

Can something like $\text{Hom}(V,K)$ be visualised?

I have no trouble visualising vector spaces like $\Bbb R^3$ and (e.g.) a subspace of dimension $2$, which would just be a plane through the origin of a $3$-D space, but I'm having trouble visualising ...
1
vote
2answers
178 views

Easy visualizations of small countable ordinals

The ordinal number $\omega^2$ can be visualized as $\omega$-many copies of $\omega$. Likewise, the ordinal number $\omega^3$ can be visualized as $\omega^2$-many copies of $\omega$, arranged as ...
7
votes
3answers
403 views

Reconciling 'intersecting planes' and 'linear transformation' interpretations of matrices

I've learned in linear algebra class that an $n \times m$ augmented matrix can be thought of as a collection of n planes in $\mathbb {R}^m$ . If the matrix is invertible, the planes all intersect at a ...
0
votes
0answers
202 views

Visualization of 2-dimensional function spaces

As a follow-up question to what is the norm measuring in function spaces I just had an idea: How about visualizing function spaces as normal planes. What I have in mind is to have an orthogonal ...