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May
12
comment Limit of multivariable functions Do not understand solution.
You are implicitly assuming that $x \ne 0$ when you use the inequality $1/(x^2+y^2) \le 1/x^2$, since otherwise the term on the right is undefined. This is fine as long as you also say what happens when $x=0$ separately. It would be better to use the given inequality to bound the numerator as: $|x^2||y| \le |x^2+y^2||y|$ to avoid this subtlety.
Apr
27
comment finding column vectors - linear transformations
The work you gave supporting your answer suggests that you don't quite understand what is being asked for, hence my question. You've accepted a correct answer below, but without understanding why it is correct you will likely have trouble with related concepts. Do you understand why the answer should be the first column of the matrix $A$?
Apr
26
comment finding column vectors - linear transformations
Do you understand what "matrix of $L$ with respect to $S$ and $T$" means?
Mar
31
comment Proving that $P[0, 1]$, the space of all polynomials on $[0, 1]$ is not complete.
Note that you don't necessarily have to show that the given sequence converges to $e^x$, just that it doesn't converge to a polynomial, which might be simpler depending on what you know about Taylor series.
Mar
29
comment Find all points on a surface which have a tangent plane parallel to given plane - is my method correct?
Also, when coming up with $y=x-\frac23$ it seems you divided by $x+y$ at some point, meaning that doing this ignores possible points for which $x+y=0$. So, you should consider these points separately at the end, meaning use $y=-x$ and $f_x=6$ to find precise points.
Mar
29
comment Find all points on a surface which have a tangent plane parallel to given plane - is my method correct?
Using only $f_x=-f_y$ isn't enough since you're ignoring that you have actual values for each of these, 6 for the first and -6 for the second. (Again, $k$ must be $1$ as you can determine by looking at the $z$-components of the gradients.) Your solution gives points for which $f_x=-f_y$ but which don't satisfy $f_x = 6$ and $f_y = -6$ as required. Once you have $y=x-\frac23$ you can then use $f_x=6$ for solve for $x$ to narrow down your values.
Mar
29
comment Find all points on a surface which have a tangent plane parallel to given plane - is my method correct?
@HelenByrne, see my comment to your original question.
Mar
29
comment Find all points on a surface which have a tangent plane parallel to given plane - is my method correct?
It doesn't make sense for the normal vectors to be vectors in $\mathbb R^2$ as you assume when you use $f_xe_1+f_ye_2$. You can't forget about the $z$-component, meaning that you should take the gradient of the three-variable function $f(x,y,z)=x^2-y^3-2xy-z$ of which your surface is a level surface, and similarly for the given plane. (In other words, you can't move the $z$ term to the right as you did.) The resulting gradients should be parallel yes, but by comparing the $z$-coordinates you can conclude that they must be equal, as the solution you commented on below suggests.
Mar
28
comment If $f(x)=\tilde{f}(\|x\|)$ and $f$ is continuous, is $\tilde{f}$ continuous?
@quid, fair enough.
Mar
28
comment If $f(x)=\tilde{f}(\|x\|)$ and $f$ is continuous, is $\tilde{f}$ continuous?
So $\tilde f$ is defined on $[0,\infty)$? You can view it as the restriction of $f$ to the nonnegative $x_1$-axis, right?
Mar
27
comment Finding the Matrix of a Given Linear Transformation T with respect to a basis
Interesting. My guess at an interpretation was that the OP is looking at the linear transformation from the space of $2 \times 2$ upper-triangular matrices to itself given by multiplication by $A$, but perhaps your interpretation is correct.
Mar
27
comment Finding the Matrix of a Given Linear Transformation T with respect to a basis
What is $T$? What is its domain and codomain? What space is the basis you provided a basis of?
Mar
27
comment Prove $\int_{a}^{b}{f(x)dx}>0$.
Fair enough. Thanks.
Mar
27
comment Prove $\int_{a}^{b}{f(x)dx}>0$.
How do you show this without (essentially) first proving the claim in the original post?
Mar
27
comment Prove $\int_{a}^{b}{f(x)dx}>0$.
Saying that $\int_{x_0-\epsilon}^{x_0+\epsilon} f(x)\,dx > 0$ since $f(x) > 0$ for all $x \in (x_0-\epsilon,x_0+\epsilon)$ essentially relies on what the OP is asking to prove, no?
Mar
27
comment Minimal polynomial and diagonalization.
Eigenvalues don't have to be distinct in order to be diagonalizable.
Mar
25
comment proving that the set of all english words is countble.
A countable union of countable sets is also countable.
Mar
24
comment $E$ compact, real-valued $f : E \to \mathbb{R}$ continuous iff graph is compact - is real valued necessary?
Saying that $g := (x,f(x))$ is continuous requires a metric on $E \times Y$ which is compatible with the product topology. So, yes, your proof does use it. Of course, there are different possible metrics with this property, but you should be clear that you are using such a metric.
Mar
24
comment $E$ compact, real-valued $f : E \to \mathbb{R}$ continuous iff graph is compact - is real valued necessary?
No those assumptions aren't necessary, but be careful with your wording. Saying that $x$ is a polynomial doesn't make sense on an arbitrary metric space, better to say that $x \mapsto x$ is the identity function on $E$. And second, for $g := (x,f(x))$ to be continuous the metric on $E \times Y$ is clearly relevant, so you should be clear about what metric you're using.
Mar
24
comment $E$ compact, real-valued $f : E \to \mathbb{R}$ continuous iff graph is compact - is real valued necessary?
@ThomasE. A continuous bijection with compact domain and Hausdorff codomain has a continuous inverse, although this should be made clearer.