Intereting Posts

Show that $\frac{\sin x}{\cos 3x}+\frac{\sin 3x}{\cos 9x}+\frac{\sin 9x}{\cos 27x} = \frac{1}{2}\left(\tan 27x-\tan x\right)$
What is the Conjunction Normal Form of a tautology?
A compact operator is completely continuous.
Prove that $\ln x \leq x – 1$
Which $p$-adic fields contain these numbers?
What is combinatorial homotopy theory?
Multiple-choice question about the probability of a random answer to itself being correct
This integral is defined ? $\displaystyle\int_0^0\frac 1x\:dx$
Taylor series for $e^z\sin(z)$
Geometric Intuition about the relation between Clifford Algebra and Exterior Algebra
Is it true that $\dim(X) \leq \dim(X^{\ast})$ for every infinite dimentional Banach space $X$?
Eigenvalues and power of a matrix
How to show $\binom{2p}{p} \equiv 2\pmod p$?
How to show an infinite number of algebraic numbers $\alpha$ and $\beta$ for $_2F_1\left(\frac13,\frac13;\frac56;-\alpha\right)=\beta\,$?
how to find the last non-zero digit of $n$

$\dim(U_1+U_2) = \dim U_1 +\dim U_2 – \dim(U_1\cap U_2).$

I want to make sure that my intuition is correct. Suppose we have two planes $U_1,U_2$ though the origin in $\mathbb{R^3}$. Since the planes meet at the origin, they also intersect, which in this case is a one-dimensional line in $\mathbb{R^3}$. To obtain the dimension of $U_1$ and $U_2$, we add the dimensions of the planes (4), and the subtract the dimensions of the line (1), which results in (3).

*additional question(s):

- V = U⊕W then Prove that (V/W)* is isomorphic to W^0
- Jordan normal form for a characteristic polynomial $(x-a)^5$
- Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix
- Finding a specific basis for an endomorphism
- Distance from point $(1,1,1,1)$ to the subspace of $R^4$
- Find a basis for $U+W$ and $U\cap W$

Can we generalize this notion to $\mathbb{F^{n}}$?

Suppose we have an additional case where $U_1$ and $U_2$ are planes in $\mathbb{R^3}$, but $U_1 \subseteq U_2$. In this instance, $dim(U_1 + U_2) < 3$, because the first two-dimensional plane is contained in the second and as a result, the dimensions of the subspaces when summed cannot exceed two. Since both subspaces $U_1,U_2$ are two dimensional and $U_1 \subseteq U_2$, then their intersection is also two-dimensional, concluding $dim(U_1+U_2)=2+2-2 = 2$.

Is this proper intuition?

- Matrix with non-negative eigenvalues (and additional assumption)
- What are examples of two non-similar invertible matrices with same minimal and characteristic polynomial and same dimension of each eigenspace?
- Linear dependency of polynomials question
- Can $A$ be singular
- Why $\dim U+\dim U^\perp=\dim V$?
- Necessary and sufficient conditions for when spectral radius equals the largest singular value.
- Eigenvalue decomposition of $D \, A \, D$ with $A$ symmetric and $D$ diagonal
- Prove two pairs of subspaces are in the same orbit using dimension
- Computing the trace and determinant of $A+B$, given eigenvalues of $A$ and an expression for $B$
- Linear w.r.t. any measure

In the latter case, they are actually the *same* plane, so their sum is again the same plane (as they are closed under addition).

Here is another (analogous) way to think about it. Let’s start with a basis $B_0$ for $U_1\cap U_2.$ We can extend $B_0$ to a basis $B_1$ for $U_1$ and a basis $B_2$ for $U_2$. Then $B_1\cup B_2$ is a basis for $U_1+U_2,$ and $B_1\cap B_2=B_0,$ so $$\begin{align}\dim(U_1+U_2) &= |B_1\cup B_2|\\ &= |B_1|+|B_2|-|B_1\cap B_2|\\ &= |B_1|+|B_2|-|B_0|\\ &= \dim(U_1)+\dim(U_2)-\dim(U_1\cap U_2).\end{align}$$ This generalizes nicely to $\Bbb F^n$, and allows us to avoid geometric arguments that may be less sensible for an arbitrary field $\Bbb F$.

Also, it will never be the case that the intersection of two planes in space is precisely $\{0\}.$ If there were two such planes $U_1$ and $U_2,$ then we would have $$\dim(U_1+U_2)=\dim(U_1)+\dim(U_2)-\dim(U_1\cap U_2)=2+2-0=4>3=\dim(\Bbb R^3),$$ which is not possible, since $U_1+U_2$ is a subspace of $\Bbb R^3$.

- Prove that $(n-1)! \equiv -1 \pmod{n}$ iff $n$ is prime
- Reference book on measure theory
- Can someone explain Cayley's Theorem step by step?
- What's the difference between stochastic and random?
- A dynamic dice game
- Need help with the integral $\int_{0}^\infty e^{-x^{2}}x^{2n+1}dx $
- Convergence/Divergence of $\int_{0}^{1/e} \frac{\log \left(\frac{1}{x}\right)}{(\log^2 (x)-1)^{3/2}} \mathrm{dx}$
- epsilon-dense property
- Point on surface closest to a plane using Lagrange multipliers
- A sine integral $\int_0^{\infty} \left(\frac{\sin x }{x }\right)^n\,\mathrm{d}x$
- Volume form and Hausdorff measure
- Cross-ratio relations
- Interesting math books for children
- Prove that between any two roots of $f$ there exists at least one root of $g$
- Proving that Zorn's Lemma implies the axiom of choice