Articles of vector spaces

Calculate the tensor product of two vectors

Let $\{e_1, e_2\}$ and $\{f_1, f_2, f_3\}$ the canonical ordered bases of $\mathbb{R}^2$ and $\mathbb{R}^3$ respectively. Find the coordinates of $x \otimes y$ with respect to the basis $\{e_i\otimes f_j\}$ of $\mathbb{R}^2\otimes\mathbb{R}^3$ where $x = (1, 1)$ and $y = (1, -2, 1)$. Since I have that $(1, 1) = e_1 + e_2$ and $(1, […]

Existence of Standard Basis

Do all vector spaces have standard bases? I was reading a book which says that “it makes no sense talking about standard bases for any vector space”. I don’t really understand this concept. If this is right, under what conditions does a vector space have a standard basis?

Transpose of composition of functions

I am trying to find an alternative proof that $(AB)^t=B^tA^t$. I think it is possible to do by showing that: $(g \circ f)^t=f^t \circ g^t$ where f,g are linear maps between vector spaces. I am unsure how to show this, can anyone point me in the right direction?

Ideal of $\text{End}_k V$ has certain form.

Let $V$ be a finite-dimensional vector space over field $k$ and $R = \text{End}_k V$. How do I see that any left ideal of $R$ takes on the form $Rr$ for some suitable element $r \in R$?

$X$ a connected space ; if all continuous functions from $X$ to $\Bbb R$ are uniformly continuous, then X is a compact

Let $X$ be a subset of a normed vector space. $X$ is also a connected space. Show that if all continuous functions from $X$ to $\Bbb R$ are uniformly continuous, then X is a compact. Need some help ; thank you 🙂

Find a basis for $U+W$ and $U\cap W$

Let $$W = \operatorname{span}([2,1,0,1], [0,0,1,0]) \\V = \operatorname{span}([1,2,1,3], [3,1,-1,4])$$ I need to find a basis and the dimension for $U+V$ and $U\cap V$. For $U+V$ I tried: $$U+V = \{u+v|u\in U, v\in V\} = \alpha_1[2,1,0,1]+\alpha_2[0,0,1,0] + \alpha_3[1,2,1,3] + \alpha_4[3,1,-1,4]$$ Therefore I have to find if this set is linearly independent or not. If it is, […]

What does this linear algebra notation mean?

I’m trying to prove that a particular $V$ is a $\Bbb{Q}$-vector space. The question says to take the element $0_V = 1$, the function $+_V : V \times V \to V$ given by the function $[x +_V y = xy]$, and the function ${}_V: \Bbb{Q} \times V \to V$ given by $[a{}_Vx = x^a]$. This […]

Necessary condition of a vector space having only one basis?

I want to know when a vector space has only one basis.

Calculating a spread of $m$ vectors in an $n$-dimensional space

My question is regarding spreading $m$ vectors in an $n$ dimensional space such that the vectors are maximally distant from each other. For example, let us say I have a 2-D space, and 3 vectors, the maximally distant spread would look as follows: If I had 4 vectors in a 3D space it would look […]

Proving that every (possibly infinite) span contains a basis in a vector space.

The book “A First Course in Algebra” says In a finite dimensional vector space, every finite set of vectors spanning the space contains a subset that is a basis. All that is fine. But what about a span having an infinite number of vectors? Surely that too must contain the basis!! An example is $\{\overline{i},\overline{j},\overline{k}+r\overline{i}\},\forall […]