# International Winter School on Gravity and Light, Tutorial 3: Multilinear Algebra – Solutions for Exercise 1

Solutions for exercise 1 of tutorial 3 of the International Winter School on Gravity and Light. (Link to video of lecture 3.)

## Notation

On this solution sheet, I’ll speak of a vector space $(V,+,\cdot)$ over a field $K$, where $+: V\times V \rightarrow V$ is the addition and $\cdot: K \times V \rightarrow V$ is called (scalar) multiplication or S-multiplication. The field $(K, \textcolor{red}{+}, \textcolor{red}{\cdot})$ has $\textcolor{red}{+}:K\times K \rightarrow K$ as addition and $\textcolor{red}{\cdot}:K\times K \rightarrow K$ as multiplication operations. The dot is often omitted, i.e. $a \mathbf v$ is short for $a \cdot \mathbf v$, $a b$ is short for $a \textcolor{red}{\cdot} b$. (Note that the lecture dealt with real vector spaces, i.e. the field $K$ was always the set of reals $\mathbb R$.) The scalars, i.e. the elements of $K$, are denoted with normal letters $a,b$, and the vectors, i.e. the elements of $V$, are denoted with boldface letters $\mathbf u, \mathbf v, \mathbf w$.

# Exercise 1: True or false?

Tick the correct statements, but not the incorrect ones. Show all answers

a) Which statements on vector spaces are correct?

?. Commutativity of multiplication is a vector space axiom. Show answer

?. Every vector is a matrix with only one column. Show answer

?. Every linear map between vector spaces can be represented by a unique quadratic matrix. Show answer

?. Every vector space has a corresponding dual vector space. Show answer

?. The set of everywhere positive functions on $\mathbb R$ with pointwise addition and S-multiplication is a vector space. Show answer

b) What is true about tensors and their components?

?. The tensor product of two tensors is a tensor. Show answer

?. You can always reconstruct a tensor from its components and the corresponding basis. Show answer

?. The number of indices of the tensor components depends on dimension. Show answer

?. The Einstein summation convention does not apply to tensor components. Show answer

?. A change of basis does not change the tensor components. Show answer

c) Given a basis for a $d$-dimensional vector space $V$, …

?.one can find exactly $d^2$-different dual bases for the corresponding dual vector space $V^*$. Show answer

?.by removing one basis vector of the basis of $V$, a basis for a $(d - 1)$-dimensional vector space $V_1$ is obtained. Show answer

?.the continuity of a map $f : V → W$ depends on the choice of basis for the vector space $W$. Show answer

?.one can extract the components of the elements of the dual vector space $V^*$. Show answer

?.each vector of $V$ can be reconstructed from its components. Show answer