Intereting Posts

A simple question about open set
Exercise books in functional analysis
When is a Morphism between Curves a Galois Extension of Function Fields
Is the ideal generated by irreducible element in principal ideal domain maximal?
congruence issue
What is the distribution of primes modulo $n$?
Proof of the distributive law in implication?
Show that: $\lim\limits_{n\to\infty} \sqrtn = 1$
If $r$ is a primitive root of odd prime $p$, prove that $\text{ind}_r (-1) = \frac{p-1}{2}$
Steinitz exchange lemma
I want to calculate the limit of: $\lim_{x \to 0} \left(\frac{2^x+8^x}{2} \right)^\frac{1}{x} $
Could I be using proof by contradiction too much?
How to find $\max\int_{a}^{b}\left (\frac{3}{4}-x-x^2 \right )\,dx$ over all possible values of $a$ and $b$, $(a<b)$?
Find the Mean for Non-Negative Integer-Valued Random Variable
In a finite field, is there ever a homomorphism from the additive group to the multiplicative group?

I was thinking about the proof of the rank-nullity theorem and I thought about proving it as follows. I just wondered whether this proof worked?

**Lemma.** If $V$ is a finite-dimensional $F$-vector space and $U\leq V$, then $V/U$ is finite dimensional.

$\hspace{16.5mm}$ Moreover, we have that $\dim{V/U}=\dim{V}-\dim{U}$.

- Minimum and Maximum eigenvalue inequality from a positive definite matrix.
- Active and passive transformations in Linear Algebra
- Signs in the natural map $\Lambda^k V \otimes \Lambda^k V^* \to \Bbbk$
- Vector space bases without axiom of choice
- 'Free Vector Space' and 'Vector Space'
- Order of general- and special linear groups over finite fields.

**Theorem (Rank-Nullity).** If $\alpha:V\to W$ is linear with $V$ finite-dimensional, then

$$\dim{V}=\dim(\text{im }\alpha)+\dim(\ker \alpha)$$

**Proof.** By the first isomorphism theorem we have

$$V/\ker{\alpha} \cong \text{im }{\alpha}.$$

Taking dimensions and applying the lemma we get

$$\dim V – \dim(\ker\alpha)=\dim(\text{im }\alpha)$$

which on rearrangement yields the result. //

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- Determine if vectors are linearly independent
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- Tensor products over field do not commute with inverse limits?
- how to find inverse of a matrix in $\Bbb Z_5$
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- QR factorization of a special structured matrix

- The sum of the residues of a meromorphic differential form on a compact Riemann surface is zero
- Is arrow notation for vectors “not mathematically mature”?
- When is the matrix $\text{diag}(\mathbf{x}) + \mathbf{1}$ invertible?
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- Why is the Riemann sum less than the value of the integral?
- Let $f:A \rightarrow B$ and C ⊂A. Define f={b∈B: b=f(a) for some a∈C}. Prove or disprove each of the statement
- Decomposition of a homogeneous polynomial
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- What would Gauss do in this case: adding $1+\frac12+\frac13+\frac14+ \dots +\frac1{100}$?
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- Using Burnside's Lemma; understanding the intuition and theory
- Continuous function $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)) = -x$?
- How to solve $\mathrm{diag}(x) \; A \; x = \mathbf{1}$ for $x\in\mathbb{R}^n$ with $A\in\mathbb{R}^{n \times n}$?