Intereting Posts

Show that there are non-well-founded models of Zermelo Fraenkel set theory
Groups such that any nontrivial normal subgroup intersects the center nontrivially
Question regarding differentiation
Are there any open mathematical puzzles?
Integrating $I=\int\frac{x}{\sqrt{x^3(a-x)}}dx, a>0$
Does proving the following statement equate to proving the twin prime conjecture?
Advantage of accepting the axiom of choice
For what values $\alpha$ for complex z $\ln(z^{\alpha}) = \alpha \ln(z)$?
Prove rank $A^TA$ = rank $A$ for any $A_{m \times n}$
The alternating group is a normal subgroup of the symmetric group
Roots of an irreducible polynomial in a finite field
Number of Fixed Point(s) of a Differentiable Function
Prime $p$ with $p^2+8$ prime
Gentle introduction to fibre bundles and gauge connections
proving $\sum\limits_{k=1}^{n} \Bigl\lfloor{\frac{k}{a}\Bigr\rfloor} =\Bigl\lfloor{\frac{(2n+b)^{2}}{8a}\Bigr\rfloor} $

I have two subspaces:

$$W_1 = \{(x, 3x) : x\in \Bbb R \}$$ and

$$W_2 = \{(2x, 0): x\in \Bbb R \}$$

- Linear algebra doubt about the use of the word 'finite'
- How to compute the determinant of a tridiagonal matrix with constant diagonals?
- eigen decomposition of an interesting matrix
- Prove $(2, x)$ is not a free $R$-module.
- Prove rank $A^TA$ = rank $A$ for any $A_{m \times n}$
- What seems to be the minors of the Adjugate matrix $\text{adj}(A)$ of a square matrix $A$?

How do I get $W_1 + W_2$?

I tried simply adding a sample vector from each, i.e. $$ (1, 3) + (2, 0) = (3, 3)$$ but I don’t think this makes sense since this new vector doesn’t fit it $W_1$ nor $W_2$….

- Prove that if $\operatorname{rank}(T) = \operatorname{rank}(T^2)$ then $R(T) \cap N(T) = \{0\}$
- Proving $W$ is a subspace of $P_{2}$
- Pseudoinverse and norm
- Find the number of $n$ by $n$ matrices conjugate to a diagonal matrix
- Proving that if $A$ is diagonalizable with non-negative eigenvalues, then $A=B^2$ for some $B$
- Cardinality of a vector space versus the cardinality of its basis
- Prove that the equation $18x+42y=22$ has no integer solution?
- Probability Calculations on Highway
- why symmetric matrices are diagonalizable?
- Why do physicists get away with thinking of the Dirac Delta functional as a function?

$W_1+W_2$ is by definition the set of **all** vectors $w_1+w_2$ such that $w_1\in W_1$ and $w_2\in W_2$. You have $$W_1=\big\{(x,3x):x\in\Bbb R\big\}=\big\{x(1,3):x\in\Bbb R\big\}$$ and $$W_2=\big\{(2x,0):x\in\Bbb R\big\}=\big\{x(2,0):x\in\Bbb R\big\}\;,$$ so you’re looking at all vectors of the form $x(1,3)+y(2,0)$ for $x,y\in\Bbb R$. Every vector in $W_1$ can be written in this form (with $y=0$), and every vector in $W_2$ can be written in this form as well, but you can’t expect every vector in $W_1+W_2$ to belong to $W_1$ or $W_2$. In fact, this occurs if and only if one of the subspaces $W_1$ and $W_2$ is a subspace of the other.

**Note:** you must combine each vector in $W_1$ with **every** vector in $W_2$, so you need to allow the coefficients $x$ and $y$ to be different; that’s why I have $x$ and $y$ and not $x$ and $x$.

Carefully note that **for any two sets** (not only for subspaces) $S$ & $T$, $S+T=${$s+t:s\in S, t\in T$}. Thus your sample vector viz $(3,3)$ is just a single element of $W_1+W_2$. You need to accommodate all such in $W_1+W_2$. Thus what should be the general form of a vector in $W_1+W_2$? Isn’t it $(x,3x)+(2y,0)$ for $x,y\in \mathbb R$?

Yes, that is the way. You have to add *all* pairs of $W_1$ and $W_2$.

So, formally

$$W_1+W_2=\{w_1+w_2\mid w_1\in W_1\text{ and }w_2\in W_2\}.$$

For example the sum of two lines (both containing the origo) in the space is the plane they span.

Anyway, it is worth to mention, that $W_1+W_2$ is the *smallest subspace* that contains $W_1\cup W_2$.

- Formal logic systems, how do we prove theorems about them?
- Evaluate $\int_{0}^{\infty} \mathrm{e}^{-x^2-x^{-2}}\, dx$
- Proving $f$ has at least one zero inside unit disk
- What are some good ways to get children excited about math?
- Geometric Interpretation of Jacobi identity for cross product
- Is a compact simplicial complex necessarily finite?
- $4494410$ and friends
- Basis of the space of linear maps between vector spaces
- Counterexample to Fubini?
- Zeroes of a Particular Function
- Formula for a periodic sequence of 1s and -1s with period 5
- What is a quotient ring and cosets?
- Prove that this iteration cuts a rational number in two irrationals $\sum_{n=0}^\infty \frac{1}{q_n^2-p_n q_n+1}+\lim_{n \to \infty} \frac{p_n}{q_n}$
- Why is integer approximation of a function interesting?
- Why do we need an integral to prove that $\frac{22}{7} > \pi$?