Intereting Posts

On an expansion of $(1+a+a^2+\cdots+a^n)^2$
Would the answer of the square root of a square root be positive or negative?
Hard inequality $ (xy+yz+zx)\left(\frac{1}{(x+y)^2}+\frac{1}{(y+z)^2}+\frac{1}{(z+x)^2}\right)\ge\frac{9}{4} $
Is trace invariant under cyclic permutation with rectangular matrices?
The 6 generals problem
Determine the centre of a circle
Bounds for lcm$(1,\dots,n)$
give a counterexample of monoid
What is the probability of of drawing at least 1 queen, 2 kings and 3 aces in a 9 card draw of a standard 52 card deck?
A problem on limit
Evaluation of $\int\frac{(1+x^2)(2+x^2)}{(x\cos x+\sin x)^4}dx$
Why are removable discontinuities even discontinuities at all?
What does the topology on $\operatorname{Spec}(R)$ tells us about $R$?
A square integrable martingale has orthogonal increments
Finding the kernel of ring homomorphisms from rings of multivariate polynomials

I am self learning **Linear Algebra** using “Linear Algebra Done Right” a book by Sheldon Axler. And for me its important to understand the smallest thing mentioned in the book. Can someone explain to me what the author means by **OVER** R/C?

And there should be a reason he mentioned this (an importance), can someone explain me that? This is what the book says :

$\mathbb{R^n}$ is a vector space **over** $\mathbb{R}$,

and $\mathbb{C^n}$ is a vector space **over** $\mathbb{C}$

- Trace of a nilpotent matrix is zero.
- $||u||\leq ||u+av|| \Longrightarrow \langle u,v\rangle=0$
- what does ∇ (upside down triangle) symbol mean in this problem
- Rotate an area around a diagonal line.
- Adding two subspaces
- set of almost complex structures on $\mathbb R^4$ as two disjoint spheres

- if $A^2 \in M_{3}(\mathbb{R})$ is diagonalizable then so is $A$
- The calculation of $\dim(U + V + W)$
- What does it mean to represent a number in term of a $2\times2$ matrix?
- How to prove that $A$ is positive semi-definite if all principal minors are non-negative?
- For which ${n\in{\Bbb Z}}$ does there exist a matrix $P\in{\Bbb C}^{4\times 4}$ such that $P^n=M$?
- Derivative of conjugate transpose of matrix
- Sign of det(UV) in SVD
- Orthogonal Projection onto the $ {L}_{2} $ Unit Ball
- Why does Friedberg say that the role of the determinant is less central than in former times?
- Invariant Subspace of Two Linear Involutions

One of the ingredients of a vector space is a definition of scalar multiplication; but you need to know what field the scalars are in! A vector space $V$ over $\mathbb{F}$ has as part of its data a map (scalar multiplication) $\mathbb{F}\times V\to V$. The same set $V$ could be given a different vector space structure with a multiplication map $\mathbb{K}\times V$; this is most common when $\mathbb{K}\subset\mathbb{F}$, and the second map is simply a restriction of the first one.

For example, $\mathbb{C}$ is a vector space over $\mathbb{C}$ as you mention in the question, but it is also a vector space over $\mathbb{R}$, since you can simply restrict scalar multiplication to real scalars. These two structures are quite different; for example, the single element $1$ is a basis of $\mathbb{C}$ over $\mathbb{C}$, since every complex number is of the form $z\cdot 1$ for some unique $z\in\mathbb{C}$. Over $\mathbb{R}$, one possible basis is $1,i$, since every complex number is of the form $a+bi$ for a unique pair $(a,b)\in\mathbb{R}^2$. Notice that the dimension of $\mathbb{C}$ changed depending on whether we gave it the structure of a vector space over $\mathbb{C}$ or over $\mathbb{R}$.

- Product of spheres embeds in Euclidean space of 1 dimension higher
- Chinese Remainder Theorem Problem
- Inclusion-Exclusion Convergence
- Power series summation
- dimension of tensor products over a submodule
- Recursive Function – $f(n)=f(an)+f(bn)+n$
- Chance of selecting the last k pages in correct order from a set of n pages
- A $\{0,1\}$-matrix with positive spectrum must have all eigenvalues equal to $1$
- How to prove that the problem cannot be solved by the four Arithmetic Operations?
- Why is $n_1 \sqrt{2} +n_2 \sqrt{3} + n_3 \sqrt{5} + n_4 \sqrt{7} $ never zero?
- Conjecture $\sum_{n=1}^\infty\frac{\ln(n+2)}{n\,(n+1)}\,\stackrel{\color{gray}?}=\,{\large\int}_0^1\frac{x\,(\ln x-1)}{\ln(1-x)}\,dx$
- Resulting equation by translating or rotating a graph.
- What's umbral calculus about?
- Random walk on vertices of a cube
- Average distance between two random points on a square with sides of length $1$