Intereting Posts

Assume that $ 1a_1+2a_2+\cdots+na_n=1$, where the $a_j$ are real numbers.
Chain rule for multiple variables?
Proving that the smooth, compactly supported functions are dense in $L^2$.
Maximum value of a function with condition
To compute $\frac{1}{2\pi i}\int_\mathcal{C} |1+z+z^2|^2 dz$ where $\mathcal{C}$ is the unit circle in $\mathbb{C}$
Claim: $a$ has $90 \% $ primes less than $n$ If $n!= 2^s \times a \times b $ and $\lfloor{\frac{a}{b}}\rfloor = 2^{s-2}$
Describe the units in $\mathbb{Z}$
Every right principal ideal non-emptily intersects the center — what is that?
Graph Theory: Forests
Topological Vector Space: $\dim Z\text{ finite}\implies Z\text{ closed}$
Statistical Inference and Manifolds
How to draw all nonisomorphic trees with n vertices?
What is the meaning of the double turnstile symbol ($\models$)?
Prove that $\sum\limits_{j=k}^n\,(-1)^{j-k}\,\binom{j}{k}\,\binom{2n-j}{j}\,2^{2(n-j)}=\binom{2n+1}{2k+1}$.
Dyson-expansion like multidimensional integral

$V = U_1\oplus U_2~\oplus~…~ \oplus~ U_n~(\dim V < ∞)$ $\implies \dim V = \dim U_1 + \dim U_2 + … + \dim U_n.$ [Using the result if $B_i$ is a basis of $U_i$ then $\cup_{i=1}^n B_i$ is a basis of $V$]

Then it suffices to show $U_i\cap U_j-\{0\}=\emptyset$ for $i\ne j.$ If not, let $v\in U_i\cap U_j-\{0\}.$ Then

\begin{align*}

v=&0\,(\in U_1)+0\,(\in U_2)\,+\ldots+0\,(\in U_{i-1})+v\,(\in U_{i})+0\,(\in U_{i+1})+\ldots\\

& +\,0\,(\in U_j)+\ldots+0\,(\in U_{n})\\

=&0\,(\in U_1)+0\,(\in U_2)+\ldots+0\,(\in U_i)+\ldots+0\,(\in U_{j-1})+\,v(\in U_{j})\\

& +\,0\,(\in U_{j+1})+\ldots+0\,(\in U_{n}).

\end{align*}

Hence $v$ fails to have a unique linear sum of elements of $U_i’s.$ Hence etc …

Am I right?

- A question about the vector space spanned by shifts of a given function
- Reflection across the plane
- Dimension of the space of algebraic Riemann curvature tensors
- Show that $T$ is normal
- How do I see that every left ideal of a square matrix ring over a field is principal?
- Visualizing the four subspaces of a matrix

- Calculate determinant of Vandermonde using specified steps.
- Linear independence of functions
- Why are $3D$ transformation matrices $4 \times 4$ instead of $3 \times 3$?
- Cauchy-Schwarz Inequality and Linear Dependence
- On the proof: $\exp(A)\exp(B)=\exp(A+B)$ , where uses the hypothesis $AB=BA$?
- Understanding Gauss-Jordan elimination
- Multiplicity of eigenvalues
- Concerning $f(x_1, \dots , x_n)$
- Isomorphism between $V$ and $V^{**}$
- Finding a closed form for a recurrence relation.

Yes, you’re correct.

Were you second guessing yourself? If so, no need to:

You’re argument is “spot on”.

If you’d like to save yourself a little space, and work, you can write your sum as:

$$ \dim V = \sum_{i = 1}^n \dim U_i$$

“…If not, let $v\in U_i\cap U_j-\{0\}.$ Then

$$v= v(\in U_i) + \sum_{\large 1\leq j\leq n; \,j\neq i} 0(\in U_j)$$

- Help on the relationship of a basis and a dual basis
- Another point of view that $\mathbb{Z}/m\mathbb{Z} \times\mathbb{Z}/n\mathbb{Z}$ is cyclic.
- Can every positive real be written as the sum of a subsequence of dot dot dot
- Show that for any subset $C\subseteq Y$, one has $f^{-1}(Y\setminus C) = X \setminus f^{-1}(C)$
- Cancellation Law for External direct product
- This sequence $\lfloor \sqrt{2003}\cdot n\rfloor $ contains an infinite number of square numbers
- Seeking proof for the formula relating Pi with its convergents
- Prove that $F(x,y)=f(x-y)$ is Borel measurable
- Equivalent Norms on $\mathbb{R}^d$ and a contraction
- For each $y \in \mathbb{R}$ either no $x$ with $f(x) = y$ or two such values of $x$. Show that $f$ is discontinuous.
- How to write permutations as product of disjoint cycles and transpositions
- Tensors which are symmetric and antisymmetric in overlapping groups
- Uniform continuity of $f(x) = x \sin{\frac{1}{x}}$ for $x \neq 0$ and $f(0) = 0.$
- Product of the logarithms of primes
- Independent, Identically Distributed Random Variables