2
votes
1answer
46 views

Isomorphism implies direct sum of Kernel and Image

If $f: U \rightarrow V$ and $g: V \rightarrow W$ are linear transformations between vector spaces over a field $K$ such that $ g \circ f$ is an isomorphism, then $V = \operatorname{Im}f \oplus ...
1
vote
0answers
36 views

$T (x_1,x_2,x_3,…,x_n) = (-x_3,x_3,x_4,x_5,…) $ then $ W \ne ker T$

Let $V$ the vector space of all sequences of real numbers and $W$ the subspace given by $W = \{(a,a,0,0,...) | a \in R\}$ , and $T : V \rightarrow V$ given by $T (x_1,x_2,x_3,...,x_n) = ...
1
vote
2answers
28 views

If $f\in V$ of degree $n$ then for every $g \in P_n(\Bbb R)$ there exist scalars $c_0,c_1,..,c_n$ such that $g = c_0f + c_1f'+ … + c_nf^{(n)}$

Let $V=P(\Bbb R)$ and $1 ≤ i$ be the vector space of the polynomials with real coefficients, on the field of real numbers $\Bbb R$. Let $T_i(f)=f^{(i)}$ the $i$th derivate of $f$. a) I have to show ...
1
vote
4answers
83 views

$\{ v_1,v_2,…,v_n\}$ is basis of $V$ if and only if $\{ v_1,v_1 + v_2,…,v_1 + v_2+…+v_n,\}$ is a basis of $V$

Let $V$ a vector space over a field $K$. Is it true $\{ v_1,v_2,...,v_n\}$ is basis of $V$ if and only if $\{ v_1,v_1 + v_2,...,v_1 + v_2+...+v_n,\}$ is a basis of $V$ ? I made some examples and ...
1
vote
0answers
31 views

Manipulating this probability distribution function

I have a probability distribution function as follows: $$ P(y|x,w, \phi) = \frac{\phi}{2\pi} \exp ^{-0.5 (y-t(x, w)'\phi (y-t(x,w)) } $$ Here $y$ and $x$ are two observed values. $\phi$ is also some ...
0
votes
1answer
14 views

Re-arrange expression to transformation form

$$\frac{6x-5}{3x+1}$$ How do you write this in the form $$\frac{b}{x+c} + a$$ I know how to find a (2) by asymptote theory, but I don't know how to re-arrange to find B.
1
vote
1answer
48 views

Inverse Function + Reflection In Y-Axis

Not getting any of the answers in MC. Is the answer wrong, or have I done something wrong?
0
votes
2answers
32 views

What formula would you use to cast an average of several numbers into a smaller range?

Say I have four numbers ranging in value from 15-95. If I want to, for a simple example, say that if the average of the four numbers is 15 (lowest possible value), that would relate to 2 on a scale of ...
1
vote
4answers
78 views

Quadratics, transformations, and formulas

Two-part question. Feel free to answer just one part, or both (write which letter part you are answering) a) If the quadratic function $g(x)=a(x-h)^2+ k$ does not touch the $x$-axis, what can be ...
2
votes
1answer
284 views

Exponential Function Shifts

I have some confusion about shifts concerning exponential functions. I can best describe my question with an example. Take y = e^-(x-3). This graph has a reflection over the y-axis and is shifted ...
2
votes
1answer
24 views

Algebraic Transformation query…

I'm boning up on Algebra, and I'm looking into Algebraic Transformation. I understand the basic concept - but I'm confused by two self assessment questions. The two questions, from what I can see, ...
2
votes
0answers
61 views

Perhaps an easy algebra problem, but it still evades me

I need help spotting a corresponding transformation Let $x,y$ be some variables and $$z=z(x,y)$$. We have a transformation $X(\lambda):(x,y,z)\to (x',y',z')$, such that $$x'= x\exp(a\lambda)\\ ...
-3
votes
1answer
71 views

Separating $x_1^{x_2}$ into sum of two terms by variable transformation

Given the function $f(x_1, x_2)$ in the form of product of two variables. $$f(x_1, x_2) = x_1^{x_2}$$ I want to apply a variable transformation on this function, so that I can write it sum or ...
2
votes
2answers
110 views

question on transformation

If a $2$d coordinate transformation function is given by $f(x,y)= 3x+1$, then what does it mean? How do I calculate the transformed coordinates for the points say $(3,4)$ in the initial space?
1
vote
1answer
51 views

Coordinate transformation to get even function

Suppose I have the function $$f(y)=2y^4-5y^3+3y^2,$$ with zeroes $y=0$ (2x), $y=1$, $y=3/2$, which I only need on the part of the domain $0\le y\le 1$. Is there a transformation $y\rightarrow y'$, ...
1
vote
1answer
329 views

Translating Cubic with Algebra?

I'm having a little trouble figuring out how to translate $ax^3+bx^2+cx+d=y$ by vector $(1,1)$ using only algebra. If possible could someone give me a hand? An example: Translate ...
0
votes
2answers
129 views

Transform/approximate this expression to avoid undefined value

I have an expression like this $$\sum_i^n\log\frac{x_i}{y_i}+\alpha\sum_i^nx_i\log\frac{x_i}{\beta}$$ A potential problem is that $x_i$ and $y_i$ may take value $0$ for certain $i$, hence making ...
1
vote
2answers
241 views

Graphing Transformations. Why does the +2 in $f(x) = \sqrt{-x + 2}$ not work as expected when done out of order?

Graphing $f(x) = \sqrt{-x + 2}$ from the graph of $y=\sqrt{x}$. Correct Method First graph $f(x) = \sqrt{x}$. then $f(x) = \sqrt{x+2}$ (shift left 2) then $f(x) = \sqrt{-x+2}$ (Reflect over ...
1
vote
2answers
121 views

Arithmetic error when calculating inverse of the logistic?

I would like to rearrange the logistic function: $$y=\frac1{1+\exp(-a+bx)}$$ To calculate $x=f(y)$ So I did the following: $$\frac1{y}=1+\exp(-a+bx)$$ $$\ln\left(\frac1{y}-1\right)=-a+bx$$ ...