Definition of sum of subspaces.

You are right that if you directly apply the definition, then $U+V = \{(x+y,y,0):x,y\in F\}$. However, $\{(x,y,0):x,y\in F\}$ and $\{(x+y,y,0):x,y\in F\}$ are the same set! Given any $(x,y,0)$ in the first set, if we let $z=x-y$ then $(x,y,0)=(z+y,y,0)$ so it is also in the second set. And given any $(x+y,y,0)$ in the second set, if we let $z=x+y$ then $(x+y,y,0)=(z,y,0)$ so it is also in the first set.

Tags:

Linear Algebra

Vector Spaces

Related

Continuity on paths implies continuity on space? Is this a Taylor series? $\ln(x) +1 = \sum_{n=0}^{\infty} \frac{n+1}{n!} \cdot \frac{(\ln(x))^n}{x}$ distance from the centre of a $n$-cube as $n \rightarrow \infty$ Solve the equation $2x^2-[x]-1=0$ where $[x]$ is biggest integer not greater than $x$. Complex quintic equation The probability that A hits a target is $\frac14$ and that of B is $\frac13$. If they fire at once and one hits the target, find $P(\text{A hits})$ Doesn't the unprovability of the continuum hypothesis prove the continuum hypothesis? Evaluate $\lim_{(x,y)\to(0,0)}\frac{\cos(xy)-1}{x^2y^2}$ $ \bigcap\limits_{n=1}^N I_n \neq \emptyset$ for all $ N \in \mathbb{N}$ implies that $ \bigcap\limits_{n=1}^\infty I_n \neq \emptyset $? An explanation for undergraduated students about why the Jacobian conjecture is hard Prove $\rm AB = BA = 0$ if the set of nonzero eigenvalues of $\rm A + B$ is union of set of nonzero eigenvalues of $\rm A$ and $\rm B$. $(\Bbb Q,+)$ is not isomorphic to any of its proper subgroups.

Recent Posts