Comment by philmcole on A symmetric matrix $A$ has eigenvalues 1 and 2. Find...
A plane orthogonal to a line can be anywhere on that line (far away from zero or near). So how do I know whichever plane I find that this is in fact the eigenspace?
View ArticleComment by philmcole on A matrix is symmetric iff its eigenspaces are orthogonal
What are $u_i$ and $v_i$?
View ArticleComment by philmcole on Is the product of symmetric positive semidefinite...
I guess his definition of definiteness is restricted to symmetric matrices only. And since $AB$ needn't be symmetric for two symmetric $A$ and $B$, we can't talk about the definiteness of $AB$.
View ArticleComment by philmcole on Prove the closure is closed and is contained in every...
@PrasunBiswas Thanks, this was a typo. I corrected it.
View ArticleComment by philmcole on Prove the closure is closed and is contained in every...
Clear as always. Thank you!
View ArticleComment by philmcole on Vector spaces of the same finite dimension are...
Why is the verification of the linearity tedious? Let $v=\sum \alpha_i v_i$ and $u=\sum \beta_i v_i$ and $\lambda$ a scalar, then $f(v+\lambda u) = f(\sum (\alpha_i + \lambda \beta_i) v_i) = \sum...
View ArticleComment by philmcole on How to prove that $f: ℝ^n →ℝ^m$ is differentiable at...
Can you elaborate how this can be used to prove the result?
View ArticleComment by philmcole on Find critical points of $f(x,y) = x \sin y + ax^2 +...
Thanks. So I can only proceed by guessing, for example that $x=y=0$ is a critical point?
View ArticleComment by philmcole on For piecewise $\mathcal C^1$ path there exists a...
@JanBohr Thanks. Also I need to guarantee that $\psi$ is continuously differentiable, monotonically increasing and bijective. So $\psi'(s)$ needs to be close to zero around the partition points too....
View ArticleComment by philmcole on For which $a$ is $y^2= x^3 + a$ a submanifold?
Oh, I see it now. Thanks!
View ArticleComment by philmcole on For which $a$ is $y^2= x^3 + a$ a submanifold?
related: math.stackexchange.com/questions/2400631/…
View ArticleComment by philmcole on Show that the closure of a subset is bounded if the...
@SaradominZamorakGuthix You can just use the closure of your open ball which is closed.
View ArticleComment by philmcole on Solve $| \frac{2+z}{2-z} | < 1$
Thanks, this was the trick I was searching for. I should try squaring the next time first!
View ArticleProve the solution of first order ODE is sum of homogeneous and particular...
Let $I \subseteq R$ be an interval and $f,g:I \to \mathbb R$ two functions. If there exists a solution $y_\text{part}:I \to \mathbb R$ of the ODE $y'+f(x)y=g(x)$, then the general solution is of the...
View ArticleAnswer by philmcole for How to prove the limit $\lim\limits_{x \to \infty}...
You don't need L'Hospital to show this.From the definition of $e^x$ as a limit we know that $(1+\frac{x}{n})^n \le e^x$ for all $n$. This is of course also true for $n+1$, which means$$e^x \ge...
View ArticleConvex function has unique zero
Let $f: [a,b] \to \mathbb R$ be a differentiable, convex function with $f(a) \gt 0$ and $f(b) \lt 0$. Then $f$ has a unique zero in $[a,b]$.The existence of the zero follows immediately from the...
View ArticleProof of convergence of newton method for convex function
I want to prove the convergence of the newton method for a convex function.Let $f: [a,b] \to \mathbb R$ be a differentiable, convex function with $f(a) \gt 0$ and $f(b) \lt 0$. Let $(x_n)_n$ be the...
View ArticleAnswer by philmcole for $(a_n)_n$ is Cauchy iff $(\Re(a_n))_n$ and...
As per MartinR's hint I solved it like this."$\implies$": Let $\varepsilon \gt 0$. Since $(a_n)_n$ is Cauchy, there exists $N \in \mathbb N$, such that for all $m,n \ge N$ we have $|a_m - a_n| \lt...
View Article$(a_n)_n$ is Cauchy iff $(\Re(a_n))_n$ and $(\Im(a_n))_n$ are Cauchy
Let $(a_n)_n$ be a Cauchy sequence in $\mathbb C$. I want to show$(a_n)_n$ is Cauchy $\iff$ the real part $(\Re(a_n))_n$ and the imaginary part $(\Im(a_n))_n$ are both Cauchy
View ArticleAnswer by philmcole for Prove if a power series is zero on an interval then...
Let $f(x) = \sum_{k=0}^\infty c_k x^k$ be a power series with radius of convergence $R \gt 0$ and $f(x)=0$ for all $|x| \lt S$ where $0 \lt S \lt R$, then $c_k=0$ for all $k$.Consider the $n$-th...
View ArticleAnswer by philmcole for Fundamental theorem of calculus with finitely many...
So because of the great help from Daniel I was able to solve the excercise. Because there are no other answers I will share my solution.The claim that $F$ is continuous is actually not dependent on the...
View ArticleFundamental theorem of calculus with finitely many discontinuities
Let $[a,b]$ be a compact interval and $f: [a,b] \to \mathbb R$ a Riemann integrable function with finitely many discontinuities.Then $F(x) := \int_a^x f(t) \, dt$ is continuous on $[a,b]$ and on all...
View ArticleAnswer by philmcole for Show coefficients of two power series agree where...
This is an application of the fact, that if a power series is zero on some interval, then its coefficients are all zero and its in fact the zero function. (see here, or here, and here, maybe here)Let...
View ArticleAnswer by philmcole for Can the theory of determinants be derived using the...
I also liked very much the approach Ted Shifrin used to introduce determinants. He motivated it by finding the inverse matrix of a $2 \times 2$ matrix and then interpreting the factor $ad-bc$ which...
View ArticleShow that if $\{Av_1,\ldots,Av_n\}$ is linearly independent then $A$ is...
I want to proof the following.Let $A$ be $n \times n$ and $\mathbf{v}_1,\ldots,\mathbf{v}_n \in \Bbb R^n$. Show that, if $\{A\mathbf{v}_1,\ldots,A\mathbf{v}_n\}$ is linearly independent, then $A$ is...
View ArticleColumn space and null space are equal for nilpotent $n \times n$ matrix with...
How can I proveLet $A$ be $n \times n$ with $n$ even, $\text{rank}(A)=n/2$ and $A^2=0$. Then $C(A) = N(A)$.I thought about proving it by showing that each set is contained in the other. Indeed the one...
View ArticleMatrix with only real eigenvalues is similar to upper triangular matrix
I want to showLet $A$ be a $n \times n$ matrix with only real eigenvalues. Then there is a basis of $\Bbb R^n$ with respect to which $A$ becomes upper triangular.There is a hint which says: Construct a...
View ArticleRestriction of diagonalizable transformation to invariant subspace is...
Let $V$ be a vector space with $\text{dim}(V)=n$ and $T: V \to V$ be a linear transformation. Let $W \subseteq V$ be a $T$-invariant subspace, i.e. $T(W) \subseteq W$, with $\text{dim}(W)=k$.Show, that...
View ArticleSymmetric linear transformation has symmetric matrix
I want show thatLet $T: \Bbb R^n \to \Bbb R^n$ be a symmetric linear transformation, i.e. $T(\mathbf{x}) \cdot \mathbf{y} = \mathbf{x} \cdot T(\mathbf{y})$ for all $\mathbf{x},\mathbf{y} \in \Bbb R^n$....
View ArticleA symmetric matrix $A$ has eigenvalues 1 and 2. Find A if $(1, 1, 1)^T$ spans...
Let $A$ be the symmetric $3 \times 3$ matrix which has eigenvalues $1$ and $2$ and $E_2 = \text{span}\left(\begin{bmatrix} 1\\1\\1\end{bmatrix}\right)$. Find $A$.What I have so far:Let $\lambda_1=1$...
View ArticleFind critical points of $f(x,y) = x \sin y + ax^2 + by^2$
I want to find all critical points of the function $f\colon \Bbb R^2 \to \Bbb R$ with $f(x,y) = x \sin y + ax^2 + by^2$ in dependence of $a,b \in \Bbb R$.I computed$$D_{(x,y)}f = \begin{bmatrix} \sin...
View ArticleIntegral over set given by $x^2+y^2 \le 1, \frac {1}{\sqrt{3}} \le...
I want to compute the integral$$\int_V x^2yz \, dx \, dy \, dz$$over the set$$V = \left\{ (x,y,z) \in \mathbb{R}^3_{>0} \mid x^2+y^2 \le 1, \frac {1}{\sqrt{3}} \le \frac{y}{x} \le \sqrt 3, z \le 1...
View ArticleVisualizing a vector field
This maybe a silly question, but I have never seen anyone rigorously define how to draw a vector field. In contrast, I have a pretty clear understanding how to draw the graph of a function.Let $f: \Bbb...
View ArticleAnswer by philmcole for How to integrate $\int{\frac{1}{\cos(x)}}dx$ using...
Here is yet another alternative.As already shown in other answers we can compute $\int \frac {1}{\cos(t)} \thinspace {\rm {d}} t$ by computing $\int \frac {1}{\sin(x)} \thinspace {\rm {d}} x$ upon...
View ArticleSolve $| \frac{2+z}{2-z} | < 1$
How do I solve this equation $$\left| \frac{2+z}{2-z} \right| < 1$$ for complex $z \in \Bbb C$?I know the answer is $\text{Re}(z) \lt 0$ but I can not understand how to get there. I tried making the...
View ArticleGeometric interpretation of rank-$1$ matrices
I need help with the following excercise about rank-$1$ matrices and their geometric interpretation. I think I managed to show the analytic parts but I struggle with the geometric interpretation parts....
View ArticleProve piecewise monotone function is Riemann integrable
I want to show the following propositionLet $[a,b]$ be a compact interval and $f: [a,b] \to \mathbb R$ be a bounded, piecewise monotone function. Then $f$ is Riemann integrable.Our definition of...
View ArticleShow the equivalence of arc length definitions
Definition 1:Let $r: [a,b] \to \Bbb R^d$ be a continuous differentiable function. Then the arc length is given by $$L(r) = \int_a^b || r'(t) || \, dt$$Definition 2:Let $r: [a,b] \to \Bbb R^d$ be a...
View ArticleProof limit and integral of sequence of continuous functions interchangeable
I want to proof the following theorem.Let $f_n: \Omega \subset \mathbb{R} \to \mathbb{R} $ be a sequence of continuous functions, $ [a,b] \subset \Omega \,$, $f_n \to f $ uniformly convergent on...
View ArticleCan a linear system $Ax=b_i$ have no, one, and infinitely many solutions for...
I am stuck on the following question from Shifrin's book Multivariable Mathematics for a while.Let $A$ be an arbitrary $m \times n$ matrix. Can there be vectors $b_1,b_2,b_3 \in R^m$, such that $A...
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