# Tagged Questions

For questions about vector spaces and their properties. More general questions about linear algebra belong under the [tag:linear-algebra] tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars. In other words, these are the spaces ...

40 views

### How to show not closed under addition/closed under addition? (Specific exam question)

I need help in part (i) and the part where I am to show that $V_3$ is a linear space. I understand that I am required to show that, in part (i), $V_1$ ∪ $V_2$ is not closed under addition; I would ...
34 views

43 views

30 views

### Proving that if T∈Hom[V,W] has null space X∩Y, then T[X+Y]=T[X]⊕T[Y]

Here's my progress: If X and Y are subspaces of a finite-dimensional vector-space, then d(X+Y)+d(X∩Y)=d(X)+d(Y) , where d(A) is the dimension of A. Then, d(X∩Y)=d(X)+d(Y)-d(X+Y). But X∩Y is ...
21 views

24 views

### How to find the base of union of 2 subspaces

Suppose we have 2 subspaces V,W. What is the base and the dimention of: V U W ? Its clear to me that V U W is not always a subspace! I was thinking about taking the base of U and the base of W and ...
53 views

### Convert velocity vector from body frame to world frame

I need to convert a 3D speed vector from a body frame to a world frame (ECEF). The velocity 3D vector I have is a linear body velocity, and I have the orientation of my object in radians in the world ...
62 views

### What is the point of “seeing” a set of polynomials or functions as a vector space?

I just had a course in linear algebra. It seemed that the main purpose is to lay the foundations of vector spaces, show ways of solving systems of linear equations and in the end, classify some ...
45 views

### On the difference between $\textbf{R}^{\{1,2,…,n\}}$, $\textbf{R}^{\{1,2,…,n+1\}}$, $\textbf{R}^{[0, 1]}$, and $\textbf{R}^\infty$

I'm working my way through Axler's "Linear Algebra Done Right" (3rd ed.), and I'm getting stuck on section 1.23, which says: If $S$ is a set, then $\textbf{F}^S$ denotes the set of functions ...
34 views

### Summation functional on a Hamel basis

Let $X$ be an infinite-dimensional Banach space. Is it possible to choose a Hamel basis $B$ of $X$ such that the linear functional defined by $f(b)=1$ ($b\in B$) was continuous?
42 views

### Vector spaces as bimodules

The usual definition of a vector space $V$ over $K$ is as an abelian group, on which $(K\setminus\{0\},\cdot)$ acts on the left, such that furthermore the operation of $K$ on $V$ is compatible with ...
21 views

19 views

### difference between linear map basis and vector basis

A linear map can be represented as a matrix in a certain basis P. Similarly, given a vector space over a field, its basis can be found, say Q. How is the concept of P related to that of Q? Are they ...
54 views

60 views

175 views

### Can someone explain this: “the set of subspaces of a vector space ordered by inclusion”

This is a claim on Wikipedia https://en.wikipedia.org/wiki/Partially_ordered_set I am not sure how to make sense of the claim What does it mean by ordered by inclusion? Inclusion as in $\subseteq$? ...
39 views

### Vector space complement to a multiplicatively closed subspace is an ideal

Let $V$ be a vector space over $\mathbb{C}$ of any dimension and suppose we have an associative multiplication $V \times V \to V$ making $V$ into a commutative ring with unity. Let $V=U \oplus W$ be a ...
### The map $f$ is degenerate or non-degenerate?
Let denote by $M_{3,2}(\mathbb C)$ the space of all $(3\times2)$-matrix of complex-dimension equal $6$ with basis $(E_{1},E_{2},E_{3},E_{4},E_{5},E_{6})$. Let $f$ a $\mathbb R$-bilinear skew-...
Solve for $\bar{x}$ and $\bar{y}$ $$\bar{x}+\bar{y}=\bar{a},~~ \bar{x}\times \bar{y}=\bar{b},~~ \bar{x}.\bar{a}=1$$ Attempt: $\bar{x}+\bar{y}=\bar{a}$ dot by $\bar{a}$, we get \$1+\bar{a}.\bar{y}=|...