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1h
answered Find subsets $W$ and $V$ of $\mathbb{R}^3$ such that $\mathbb{R}(W\cap V)\neq\mathbb{R}W\cap \mathbb{R}V$.
21h
comment Injectivity of the function $x||x||$ on $\mathbb R^n$
How about, $f(x) = f(y)$ implies $\|x\| = \|y\|$ and $x/\|x\| = y/\|y\|$
21h
comment When can a set have an upper bound but no least upper bound?
@liqiudilk $0$ is an upper bound of $\emptyset$ because every element of $\emptyset$ is less than $0$.
21h
comment Injectivity of the function $x||x||$ on $\mathbb R^n$
If $f(x) = f(y)$, you need to show that $x = y$. These pieces of information are enough to do so.
21h
answered $U(\mathbb{C}^n)$, $SU(\mathbb{C}^n)$ connected subsets of $M_n(\mathbb{C})$?
21h
comment what does it by raising a matrix to the power of $1/2$?
Positive definite matrices are symmetric (Hermitian). $A^{1/2}$ is usually defined to be the positive definite square root.
21h
comment Injectivity of the function $x||x||$ on $\mathbb R^n$
@S.Panja-1729 for example, $f:\Bbb R \to \Bbb R$ given by $f(x) = e^x$ doesn't have an inverse $f^{-1}: \Bbb R \to \Bbb R$.
21h
comment Injectivity of the function $x||x||$ on $\mathbb R^n$
@ElliotG depends how you define the domain of the inverse
21h
answered Injectivity of the function $x||x||$ on $\mathbb R^n$
21h
comment Injectivity of the function $x||x||$ on $\mathbb R^n$
Your first statement alone is not enough to deduce injectivity.
21h
comment what does it by raising a matrix to the power of $1/2$?
The conjugate transpose (aka adjoint).
22h
comment what does it by raising a matrix to the power of $1/2$?
If $M$ is positive definite and $X$ is any invertible matrix, $XMX^*$ will be positive definite (and generally won't be similar). That's actually exactly what I wrote... the point then is not that $S = S^{-1}$, but that $S = S^*$.
22h
comment Why is $\frac{\partial }{\partial y}\int M dx = \int \frac{\partial M}{\partial y}dx$
See the Leibniz integral rule
22h
comment what does it by raising a matrix to the power of $1/2$?
That last term is really $$ A^{-1/2}[A(I-B)]A^{-1/2} $$ the $A$ there cancels.
1d
comment Find all functions f such that $f(f(x))=f(x)+x$
@Jason in particular, we have a lot of $\Bbb Q$-linear functions.
1d
comment Find all functions f such that $f(f(x))=f(x)+x$
@Jason it will help because (assuming the axiom of choice) there are a lot of crazy discontinuous solutions.
1d
comment Eigen values of A*A are non negatives.
A common definition of the adjoint $A^*$ (which turns out to be the conjugate-transpose for matrices) is that $A^*$ is defined to be the unique linear transformation satisfying $$ \langle Ax,y \rangle = \langle x,A^*y \rangle $$ for all vectors $x,y$.
1d
comment Find all functions f such that $f(f(x))=f(x)+x$
Do we want to suppose that $f$ is continuous?
1d
answered TRIGONOMETRICAL IDENTITIES
1d
comment Let $F$ be the set of all functions of the form $f: t\to \sum_{k=1}^n a_k \cos(kt)+b_k \sin(kt)$. Is $F$ an integral domain? Is it a field?
$f(0) = \sum a_k$, so your kernel is not equal to the set of functions under consideration here.