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5h
comment Showing that a map, $R:\mathbb{R}^n\rightarrow\mathbb{R}^n$ can be represented by an orthogonal matrix.
This matrix is not orthogonal, and you also seem to assume that $n=3$.
2d
comment True or False Linear Algebra-Subspace
Do you know the definition of "subspace"?
Sep
27
comment What is the difference between a Limit and Derivative?
A derivative is a specific type of limit, but "limit" is a much more general notion.
Sep
25
comment An estimate for the lower Riemann sum for the derivative of a differentiable function
It is not assumed that $f'$ is integrable.
Sep
22
comment Can a matrix span $\mathbb{R}^3$
It does not make sense to say that a matrix spans $\mathbb R^n$. Do you mean to ask whether the columns of this matrix do so instead?
Sep
19
comment Prove that the dual space $V^{\ast}$ has the direct-sum decomposition $V^{\ast}=V_1^0\oplus \cdots \oplus V_k^0$.
Are you using $V_i^0$ to denote the annihilators? You should make this clear. Also, what thoughts do you have on the problem?
Sep
18
comment Is it true: A real symmetric matrix is either positive definite or negative definite or indefinite?
No, the matrix given in Will Jagy's answer is none of those. You're missing the "semidefinite" possibilities.
Sep
18
comment Is it true: A real symmetric matrix is either positive definite or negative definite or indefinite?
This is not correct. See my comment above.
Sep
18
comment Is it true: A real symmetric matrix is either positive definite or negative definite or indefinite?
Is the zero matrix positive/negative definite or indefinite?
Sep
16
comment Doubts about definition of open sets in “Understanding Analysis” by Stephen Abbott
@Hunter, what's the mistake? In the definition you quote only defines "open" for subsets of $\mathbb R$, in which case his definition of $\epsilon$-neighborhood is fine.
Sep
15
awarded  Yearling
Sep
11
comment Can there be a Finite Field That Does Use Not Modular Arithmetic?
You can transport the multiplication on your $\mathbb F_4$ to $\mathbb Z_2 \times \mathbb Z_2$, thereby turning $\mathbb Z_2 \times \mathbb Z_2$ into a field. Yes, this multiplication is not "multiplication mod $p$'', but modular arithmetic is still present in the additive construction regardless. EDIT: Seems we're on the same page ;)
Sep
11
comment Can there be a Finite Field That Does Use Not Modular Arithmetic?
But the additive group on your $\mathbb F_4$ is isomorphic to $\mathbb Z_2 \times \mathbb Z_2$, so I would argue that it does use modular arithmetic. Similarly, $\mathbb F_{p^n}$ uses addition mod $p$.
Sep
10
comment Are the Cauchy-Riemann equations and the continuity of partials enough for analyticity?
@Ian, the function is the constant $1$ along the $x$-axis and along the $y$-axis, so all partials are zero at the origin.
Sep
9
comment Are the Cauchy-Riemann equations and the continuity of partials enough for analyticity?
The partial derivatives do exist at the origin, it's their nonexistence elsewhere which is the problem.
Sep
9
answered Are the Cauchy-Riemann equations and the continuity of partials enough for analyticity?
Sep
4
comment Prove that $V = \ker(\phi) \oplus \text{image}(\phi)$
Why is $V = \operatorname{ker}(\phi^n)+\operatorname{image}(\phi^n)$? This isn't true for a general linear map in place of $\phi^n$.
Aug
15
comment Infinite dimensional Vector Spaces and Bases of Quotients
How would you do this in the finite-dimensional case? Your suggestion doesn't even work there: $(1,1)$ and $(1,-1)$ form a basis of $\mathbb R^2$ but your procedure does not produce a basis of the quotient of $\mathbb R^2$ by the $x$-axis.
Aug
14
comment Prove that $T^n$ is diagonalizable.
You need to assume that the matrix is not a scalar multiple of the identity.
Aug
14
comment Prove that $T^n$ is diagonalizable.
Did you try to see what happens in specific examples? Take your favorite non-diagonalizable $2 \times 2$ matrix, is its square diagonalizable?