Tagged Questions
1
vote
2answers
34 views
Solving the domain and range of a region satisfying two inequalities?
The question I was provided was:
"Find the domain and range of the region satisfied by the following inequalities:
i) $y \ge (x-1)^2$
ii)$y \le2x+1$
Any help would be greatly appreciated. Would you ...
0
votes
1answer
47 views
Additive maps modulo $1$ - what do they look like?
Recently, some convergence problems I have been considering led me to look at additive maps of the torus (which for me is the additive group $T := \mathbb{R}/\mathbb{Z}$).
A map $f:\ T \to T$ is ...
2
votes
0answers
13 views
Repertoire method for solving recursions
I am trying to solve this four parameter recurrence from exercise 1.16 in Concrete Mathematics:
$g(1) = \alpha;$
$g(2n+j) = 3g(n) + γn + β_j$ : j = 0,1 and n >= 1
I have assumed the closed form to ...
1
vote
2answers
75 views
Prove that the only eigenvalue of a nilpotent operator is 0?
I need to prove that if $\phi : V \rightarrow V$ is nilpotent, then its only eigenvalue is 0. I know how to prove that this for a nilpotent matrix, but I'm not sure in the case of an operator. How ...
0
votes
1answer
25 views
Linear Algebra Proof of Injective Function
I'm new in the University and I don't know how to solve this:
Suppose $v$ is a non null element of a vector space $V$ on $\mathbb R$. Show that the function is injection:
$\mathbb R\to V $
$t ...
4
votes
1answer
80 views
Is this matrix function convex or non-convex?
Given, $g(Z)=Tr(Z^Tf(Z)Z)$ , where $f(Z)=h(Z)-ZZ^T$ is a p.s.d matrix formed using entries in $Z$, where again $h(Z)$ is a diagonal matrix with its $i$'th diagonal entry being ...
5
votes
2answers
85 views
Concept of Linearity
I hear so many terms involving the word "linear". Linear function, linear equation, linear system, linear operator, linear transformation, linear mapping, linear space, linear algebra, linear ...
2
votes
2answers
107 views
Find the matrix of the given linear transformation T.
Here's the specific scenario:
$T(M) = \begin{bmatrix}1&2\\0&3\end{bmatrix}M$ from $U^{2 \times 2}$ to $U^{2 \times 2}$ with respect to $\mathfrak{B} ...
0
votes
2answers
81 views
Convexity of $\log \det(I+(P-\text{tr}(X))\cdot X)$
where $P-\text{tr}(X)>0$ and $X$ is a diagonal matrix with diagonal elements are $(x_1,x_2,\dots x_m)$
4
votes
2answers
137 views
Prove that function is bijective
Let $n \in \mathbb{N} \setminus \{ 0 \} $ and $A \in M_n(\mathbb{R})$ with $m \in \mathbb{N} \setminus \{ 0 \}$ as $A^m= \alpha \times I_n$, with $ \alpha \in \mathbb{R} \setminus \{ -1,1 \}$.
...
0
votes
0answers
32 views
backward stability of full svd decomposition
Why is it impossible for the full SVD decomposition of a matrix A to be a backward stable algorithm? This was mentioned in one of my readings but it doesn't explain why.
1
vote
2answers
107 views
Describe whether the function is one to one and onto. If both, describe its inverse.
$f:R^{2}\rightarrow R^{2}$ where $f(x,y)=(2x+y,x+y)$.
So I know that the function is one to one and onto and don't need help proving that. But with its inverse, I am a little confused. My professor ...
1
vote
1answer
62 views
What's the difference between T(V) and ImT?
Assuming I have the following Linear transformation:
$\mathbb{T}: \mathbb{V} \rightarrow \mathbb{W}$ where $\mathbb {V}$ and $\mathbb {W}$ are vector space.
...
0
votes
0answers
45 views
Sets of functions and its subsets.
Im trying to show that each set is a vector space of the precceding set : $$ \mathcal{F}_P \subseteq \mathcal{F}_D \subseteq \mathcal{F}_C \subseteq \mathcal{F}_I \subseteq \mathcal{F}_B \subseteq ...
0
votes
0answers
133 views
Approximate a complicated mystery function
Let there exist a mystery function ƒ.
ƒ accepts exactly 2 arguments, A & B.
As B approaches A, ƒ approaches A, at a simple exponential growth rate E.
As B approaches 0, ƒ approaches the mean ...
4
votes
3answers
128 views
Image of function definition notation
In my Linear Algebra and Geometry textbook, it defines the image of a linear transformation $T$ as:
$$\operatorname{Im}\, (T) := \{\; w \in W : \; w=Tv \;\;\text{ for some } v \in V \} $$
As far as ...
0
votes
1answer
44 views
linear independence in $F(\mathbb R,\mathbb R)$
For each subset of each vector space, justify whether their vectors are linearly independent or dependent:
$$\{f,g,h\}, f(x)=e^{2x} , g(x)=x^2 , h(x)=x,\mbox{ in }F(\mathbb R,\mathbb R).$$
1
vote
0answers
21 views
Stochastically sampling a range of UV's
Math newbie here. I am having some difficulties figuring out how to stochastically sample a range of UV points.
example: [ (u,v) | v <- [5..-5], u <- [-7..7] ]
I would need to make sure this ...
0
votes
0answers
35 views
Extracting a function from set of inequalities
I have set of inequalities in two dimension space which represent relation between $X$ and $Y$. now I want a function whose input is $X$ and output is $Y$. In other words, I want $F$ such that ...
0
votes
0answers
51 views
How to find function coefficients
I'm not an expert in math but I need to solve the following task:
I have several functions:
$$
f(t)=k_1 f_1(t)+k_2 f_2(t)+k_3 f_3(t)+ \dotsc +k_n f_n(t)
$$
Also I know all the functions' values: ...
3
votes
2answers
79 views
Is the function linear?
Given
$$ F(x) = \left(\begin{matrix} -x_2 \\ x_1+2x_2 \\ 3x_1 - 4x_2 \end{matrix} \right), x = \left(\begin{matrix} x_1 \\ x_2 \end{matrix} \right)$$
Prove whether $F$ is a linear function or not.
...
1
vote
1answer
41 views
General solvability at the stationary condition
Suppose a convex quadratic function $f(x)$ is given. To find a minimum of such function, one sets its derivative so zero, and solves for $x$. For instance, suppose that the result of differentiation ...
5
votes
2answers
214 views
Are all multiplicative functions additive?
Suppose $cf(x)=f(cx)$ and $f:\mathbb{R}\to\mathbb{R}$. I believe it follows that $f(x+y)=f(x)+f(y)$.
Proof: There is some $c$ such that $y=cx$. Then
...
4
votes
1answer
66 views
Can a transformation matrix be expressed in terms of the vector to be transformed?
I'm currently learning linear algebra with my friend via an online course, and we have a disagreement that we would like settled.
Upon learning that vectors can be projected onto lines by a simple ...
5
votes
1answer
88 views
Are translations of logarithms linearly independent?
I think I proved the following but I am not sure. I will write my answer at the bottom
Is the set of logarithms $ \lbrace\ln (t + a_i)\rbrace_{i=1}^N $ with $t,a_i>0$, and all $a_i$ different ...
0
votes
2answers
72 views
Searching for a suitable invertible function
I am searching for a monotonically increasing and invertible function in $2$ variables. I know several monotonically increasing functions. This is also true for invertible functions. But I am ...
7
votes
3answers
793 views
What is a basis for the vector space of continuous functions?
A natural vector space is the set of continuous functions on $\mathbb{R}$. Is there a nice basis for this vector space? Or is this one of those situations where we're guaranteed a basis by invoking ...
4
votes
1answer
185 views
What is the Jacobian?
What is the Jacobian of the function $f(u+iv)={u+iv-a\over u+iv-b}$?
I think the Jacobian should be something of the form $\left(\begin{matrix}
{\partial f_1\over\partial u} & {\partial ...
1
vote
1answer
204 views
Formula To Determine Percentage Between Two Numbers After Certain Threshold
I have a formula I use to determine how opaque some validation text should be based upon the length of a user's input compared to the maximum lenth allowed. I want to modify it so that the "ramping ...
2
votes
2answers
97 views
linear maps $\mathbb{R} \to \mathbb{R}$ are bijective or zero
Proof that every linear map $\phi: \mathbb{R} \to \mathbb{R}$ is bijective or zero.
It's not true for $\mathbb{R}^n, n \geq 2$, but how to proof/argue that it is true for $\mathbb{R} \to \mathbb{R}$?
...
4
votes
2answers
370 views
surjective linear map from R to R²
How to quickly and clearly argue/show that there is or is not a linear surjective map $\phi$
$\phi: \mathbb{R} \to \mathbb{R}²$
1
vote
2answers
52 views
matrix to vector
What's the formal way to map a Matrix $A \in M(n \times n, K)$ to a row vector $B \in K^{n²}$ where
a)
the columns
$col_i(A)\quad, \quad 1 \leq i \leq n$
are arranged one below the other
...
0
votes
1answer
165 views
How to learn graph plots of math functions?
I really don't know how to we say that a log function would look like this or polynomial function would look like this. I know that if I have like $X + Y = c$, I can draw straight line by taking ...
1
vote
2answers
53 views
Linear independence of linear combination of functions
This question is inspired by this question. Is it always true that if $\{f_n(x)\}_n$ are linearly independent then so is $\{f_n(x)+f_n(-x)\}$ given that each $f_n(x)$ do not have definite parity?
0
votes
1answer
64 views
Parity, Set of functions
Given linearly independent $\{f_n\}_{n=1}^N$ how can we form $n$ linearly independent functions $\{F_n\}_{n=1}^N$ such that each $F_n$ is either an even or odd function? Thanks.
2
votes
1answer
103 views
Set of linear transformations
How to determine the dimension of {$T\in$ Lin($X,Y$) s.t. $T(A)\subset B$} where $A,B$ are subspaces of finite-dimensional vector spaces $X$ and $Y$? Thanks in advance!
P.S. Is there a general way of ...
0
votes
3answers
819 views
Converting this recursive function into a non-recursive equation
I am trying to convert the following recursive function to a non-recursive equation:
$$
f(n) =
\begin{cases}
0,&\text{if n = 0;}\newline
2 \times f(n -1) + 1,&\text{otherwise.}
...
7
votes
4answers
269 views
The leap to infinite dimensions
Extending this question, page 447 of Gilbert Strang's Algebra book says
What does it mean for a vector to have infinitely many components? There are two different answers, both good:
1) The ...
1
vote
3answers
160 views
Generating an N-Dimensional Convex Quadratic Function
I am currently working on a research project that is trying to show that some numerical integration techniques that do well on separable convex functions will not necessarily do well on non-separable ...
0
votes
1answer
136 views
Injective on $L^n$: what does it mean?
I came across the following sentence in an article:
$f$ is injective on $L^1(\mathbb{R})$
where $f$ is some function of $g$. Now I have a vague idea that injective= one-to-one, and can visualize ...
4
votes
3answers
214 views
Proof of linear independence of $e^{at}$
Given $\left\{ a_{i}\right\} _{i=0}^{n}\subset\mathbb{R}$ which are
distinct, show that $\left\{ e^{a_{i}t}\right\} \subset C^{0}\left(\mathbb{R},\mathbb{R}\right)$,
form a linearly independent set ...
1
vote
1answer
218 views
transformation matrices and complex functions as projections
This question is about the connection between linear algebra and complex analysis. Coming from a two real dimensional domain a transformation matrix geometrically transforms a set of points (e.g. a ...
