Lecture 3: Multilinear Algebra (International Winter School on Gravity and Light 2015)

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Strong rec peek-a-boo · 2021-10 · Physics Stack Exchange · on “Confusion about word "vector" in "basis vectors" of General Relativity”

I STRONGLY suggest you watch lectures 3 [this video] and 5 [LINK] from Frederic Schuller's General relativity lectures. Lecture 3 is about (multi)linear algebra (a subject which can be learnt completely independently of any general relativity), and lecture 5 is about tangent spaces, because it seems like you're mixing up several basic concepts.

Recommends Antoni Parellada · 2016-11 · Mathematics Stack Exchange · on “Understanding the double dual space in a concrete example”

I am trying to explore the assertion that in finite vector spaces $V^{**}=V.$ This statement is most likely incorrect, as stated on this accepted answer in MathSE [LINK] , although it seems clearly spelled out as true on this precise point of a youtube series by Ben Garside [LINK] , excellent in its didactic approach, or in this minute on a lecture by Frederic Schuller [this video] , part of the series Gravity and Light [LINK] . But I'd like to focus on a concrete example of a vector space presented here [this video] as defined over the set of polynomials of fixed degree $(N=7)$ on the interval $(-1,+1)$: $$\mathcal P := \{p:(-1,+1)\rightarrow \mathbb R \, \vert \, p(x)=\sum_{n=0}^N p_nx^n\}$$ So the

Recommends peek-a-boo · 2020-07 · Physics Stack Exchange · on “Are $v^ie_{i}$ and $v^iv_{i}$ (where $v$ are the components and $e$ the basis vectors) both tensors? Or only the second one?”

I suggest you watch this lecture [this video] or this one [LINK] , by Frederic Schuller, on linear and multilinear algebra to understand the relationship between tensors as multilinear maps on a vector space vs their components with respect to a particular basis (and if you're studying General relativity, then continue watching the rest of the lecture series, it is very helpful).

Recommends Ben H · 2022-12 · Mathematics Stack Exchange · on “What is the operation that happens between a tensor product and a vector under matrix multiplication?”

I would recommend Schuller's lecture on Multilinear Algebra [this video] for a very nice from-the-ground-up description of vector spaces and tensors, but it doesn't include a discussion of the tensor product. Lee's Riemannian Manifolds has a very brief review of tensors in Chapter 2 [LINK] , which does include tensor products. The essentials are: Given a $d$ -dimensional vector space $V$ over the reals, the dual space of covectors $V^*$ is the set of all linear maps from the vector space to the reals, i.e., $\omega \in V^*$ means: \begin{align} \omega: V &\rightarrow \mathbb{R}\\ v &\mapsto \omega(v) \in \mathbb{R} \end{align} An $(l, k)$ -tensor is a linear map from $l$ copies of $V^*$ and $k$ c

Recommends Ben H · 2022-12 · Mathematics Stack Exchange · on “What is the operation that happens between a tensor product and a vector under matrix multiplication?”

See, for example this section of Schuller's GR tutorials [LINK] , where a $(1,1)$-tensor is used to create a map from a co-vector to a co-vector (instead of a map from a vector to a vector). And, of course, watch the associated video on Multilinear Algebra [this video] by Schuller.

Recommends Ben H · 2023-02 · Physics Stack Exchange · on “What is the connection between a mathematician and physicist's definition of a tensor?”

Most physicists working in general relativity call the components of a tensor, a tensor. Then they have requirements for the transformation of these components under change of basis to guarantee that these indeed satisfy the properties of the components of an (abstract) tensor. Frederic Schuller's lectures on GR [LINK] give a very nice presentation from a mathematical perspective, but he makes the connections to the physicist's notation/shorthand at a number of places, e.g., "and now comes something everybody has been waiting for who knew tensors before, namely components of tensors" [this video] and "... so this makes GR accessible to the masses" [LINK]

Recommends Ben H · 2024-03 · Physics Stack Exchange · on “In general relativity, is gauge invariance the same as coordinate invariance?”

Yes it is something that the physics description of General Relativity misses (or at least does not keep at the front of mind). Choosing a coordinates means to choose a basis of vectors at every point, like $\left\{ \frac{\partial}{\partial x^\mu} \right\}$. Then, the metric tensor, which takes two vectors and gives back a scalar, $g: V \times V \rightarrow \mathbb{R}$, will have components, $g_{\mu \nu} := g\left(\frac{\partial}{\partial x^\mu}, \frac{\partial}{\partial x^\nu}\right)$. Check out Frederic Schuller's Multilinear Algebra lecture [this video] .

Recommends Ben H · 2024-03 · Physics Stack Exchange · on “When determining the dual basis do you use the rotation matrix to rotate the basis vectors by 90 degrees?”

Do yourself a favor: use the lectures on relativity by Frederic Schuller [LINK] . Lecture 3 is on Multilinear Algebra [this video] .

Cites Charlie · 2019-12 · Physics Stack Exchange · on “Correct way of viewing a (1,1)-tensor returning a vector”

I am currently watching this [this video] excellent video series building up to general relativity. We have finally started looking at tensors and a question came up from the audience which (to my understanding) was asking why tensors are defined as multi-linear maps from sets of vectors and covectors to the real numbers yet (1,1)-tensors are often viewed as linear maps from vectors to vectors. He says that the two ways of viewing the (1,1)-tensor contain the same information, but I don't understand how. If anyone could provide another explanation for why this is true I would really appreciate it.

Cites Taylor Rendon · 2020-05 · Mathematics Stack Exchange · on “What is meant by “how an element in the domain is mapped to its image”.”

In the following lecture [this video] given by Fredric Schuller, he mentions this during the lecture which is on multilinear algebra (know that $P$ is a set of polynomials such that $p$ $\in$ $P$ ): “consider the map $I$ : $P$ $\to$ $\mathbb{R}$ , now I need to say how a polynomial is mapped to a real number $I(p)$ $:= $$\int_0^1$$p(x)dx$ “ My questions is: What does he mean when he says “how a polynomial is mapped to a real number”? Is he simply referring to the definition of $I$ ? I’ve just never heard someone say that instead of saying “The map $I$ is defined as...”