Intereting Posts

The range of a continuous function on the order topology is convex
Sine values being rational
Optimal multiplying method
How to prove that $e = \lim_{n \to \infty} (\sqrt{n})^{\pi(n)} = \lim_{n \to \infty} \sqrt{n\#} $?
Trouble with an Inequality
Galois group command for Magma online calculator?
If a coin toss is observed to come up as heads many times, does that affect the probability of the next toss?
Formula about time derivative of pushforward of family of forms: where is it from?
Infinite Lebesgue integral over all infinite measure subsets?
Computing Residues
Examples of statements which are true but not provable
What is the integral of $\int e^x\,\sin x\,\,dx$?
Why is the Koch curve homeomorphic to $$?
Problem about jointly continuous and linearity of expectation.
Find $\det X$ if $8GX=XX^T$

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** $\mathbb{C}$

- Minimal polynomial of $A := \left(\begin{smallmatrix} 7 & -2 & 1 \\ -2 & 10 & -2 \\ 1 & -2 & 7 \end{smallmatrix}\right)$
- What does double vertical-line means in linear algebra?
- Multiplicity of eigenvalues
- Is adjoint of singular matrix singular? What would be its rank?
- If $A$ is a non-square matrix with orthonormal columns, what is $A^+$?
- Eigenvalues of a certain tridiagonal matrix

- Prove that for a real matrix $A$, $\ker(A) = \ker(A^TA)$
- Motivation for spectral graph theory.
- How to solve this equation? $x^2 - y^2 = 5$ & $ xy = 2$. Find $x$ and $y.$
- Etymology of the word “normal” (perpendicular)
- Uniqueness of the Jordan decomposition.
- Diagonalisable or not?
- A method of finding the eigenvector that I don't fully understand
- Find the real vector $x$ which satisfies all this?
- geometric multiplicity= algebraic multiplicity for a symmetric matrix
- Let $(P_3)_E$ denote the space $P_3$ with the standard basis $E = \{1, x, x^2\}$.

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 $\mathbb{R}$.

- $\lfloor a n\rfloor \lfloor b n\rfloor \lfloor c n\rfloor = \lfloor d n\rfloor \lfloor e n\rfloor \lfloor f n\rfloor$ for all $n$
- A Tale of Two Quadratic Identities (Pell-like)
- Convergence of the sequence $\frac{1}{n\sin(n)}$
- Tetrahedral Law of Cosines Proof
- Compact sets in R
- Abstract Algebra
- Proving $\prod((k^2-1)/k^2)=(n+1)/(2n)$ by induction
- If $A$ is positive definite, then $\int_{\mathbb{R}^n}\mathrm{e}^{-\langle Ax,x\rangle}\text{d}x=\left|\det\left({\pi}^{-1}A\right)\right|^{-1/2}$
- How to interpret Fourier Transform result?
- Find $\lim_{x\to 1}\frac{p}{1-x^p}-\frac{q}{1-x^q}$
- Why does $\mathbb{R}$ have the same cardinality as $\mathcal{P}(\mathbb{N})$?
- How smooth can non-nice associative operations on the reals be?
- Solutions of $f(f(z)) = e^z$
- $S\subset \mathbb{R}^2$ with one and only one limit point in $\mathbb{R}^2$ such that no three points in $S$ are collinear
- a q-continued fraction related to the octahedral group