1
vote
0answers
17 views

finding the symmetric point

let there be $4$ points. $A(-1,1,1), B(2,0,-1), C(1,3,-2), D(-2,-1,0)$. the $4$ points are not on the same line. the plane which goes through the points $A$ and $B$, and which is also paralel to the ...
0
votes
0answers
52 views

Solve system of equations for the ratios of the vectors

(Sorry for the bad title, didn't think of a better way to describe the problem). I have a system $\mathbf{A}\in\mathbb{C}$ that forms the problem $\mathbf{Ax}=\mathbf{b}$, for which I want to find an ...
0
votes
1answer
20 views

Graph exponential function

I am having problems understanding why $xe^x + 10e^x$ has two $(x,y)$ intercepts. I understand why there is one $(0,10)$, but am unclear on how to return $(-10,0)$. Any help would be much ...
0
votes
1answer
26 views

Finite sum equaling Kronecker Delta

could anyone help understand how $$\sum_{j=0}^{n-r}\binom{n-r}{j}*(-1)^{j} = [1 + (-1)]^{n-r}$$ I see that if $j=0$, i get $1=1^{n-r}$, and if $j=n-r$, i get $(-1)^{n-r},$ but what about the rest of ...
1
vote
1answer
72 views

How to solve this graphing question?

$ \frac{|x-2|} {(x^2-4)}+\frac{(x-2)} {|x-2|} = b $ determine for which values of $b$ the equation has one and only solution. I tried sketching the graph, but was unable to do so accuratly...also, ...
0
votes
1answer
23 views

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept.

For the polynomial, list each real zero and its multiplicity. Determine whether the graph crosses or touches the x-axis at each x -intercept. f(x) = (1/5)x^4(x^2 - 3) the choice 1- 0, ...
0
votes
0answers
31 views

Using Algebra with Trig Functions

Using Algebra with Trig Functions I'm trying to find the correct 1 second audio signal I would need to apply to a 1 second known noise signal to have the output signal be a sin wave. The basic ...
0
votes
4answers
60 views

How can I factor $x^2 + 2\sqrt{3}\,x + 3$? [closed]

$$x^2 + 2\sqrt{3}\,x + 3$$ Anyone could tell me how may I factor this? Thanks a lot
0
votes
4answers
46 views

algebraic representation of a line in 3d

Is an algebraic representation of a line in 3d possible, or there can be only a parametric one?
0
votes
1answer
20 views

the volume of pyramid value

when calculating the volume of pyramid using a determinnat, is it ok to take the determinanat in absloute value so that every negative result would be converted to positive volume number?
2
votes
1answer
44 views

Vector calculation question

the points a b c d are concordantly ( 1,2,-3) , (-1,2,1) , ( 0,1,-2) , ( 2,-1,1) find formula of the plane going thorugh d and which is pararlel to plane abc calculate the volume of pyramid abcd. ...
0
votes
2answers
55 views

explain this confusing algebraic identity?

Can anyone show, step-by-step, how the expression on the LHS can be turned into the expression on the RHS? $x^ay^b=a^ab^b(a+b)^{-(a+b)}(x+y)^{a+b}$
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
81 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 ...
3
votes
2answers
168 views

Linear algebra calculus trick.

I have a matrix and a vector: $$ A=\begin{bmatrix} a &b\\ c&d \end{bmatrix}, $$ $$ \vec v=\begin{bmatrix} a+b\\ c+d \end{bmatrix} $$ Is there an algebraic operation that produce the ...
-1
votes
0answers
17 views

Linear Equation Formulas for specific questions

Im trying to figure out how to do this problem, but it is just extremely confusing to understand how too do. A cricket chirps at different retes depending on temperature. You can estimate the ...
0
votes
0answers
12 views

Interpreting & Analysing a Transitional Matrix

How do you interpret such a problem Are we expect to add the rows, and that would be the one with larger number of goats in the long term. Therefore A(row 1) and b(row 2)... therefore the answer is ...
0
votes
3answers
40 views

Explanation on characterstic polynomial

$A_2 = \begin{pmatrix} 1 & 1 \\ a & 1 \end{pmatrix} $ So the characteristic polynomial of $A_2$ is $P_a(t) = (t-1)^2 - a $ Then, $ P_a(t) = t^2 -2t +1 -a$ ...
0
votes
3answers
42 views

What am I doing wrong in searching for the intersection point between 2 linear equations?

y=4x+5 y=3x-7 First I take equation 1 and set y to 0: 0 = 4x + 5 -5 = 4x -5/4 = x I get that the information above only ...
0
votes
4answers
240 views

Finding an equation of circle which passes through three points

How to find the equation of a circle which passes through these points $(5,10), (-5,0),(9,-6)$ using the formula $(x-q)^2 + (y-p)^2 = r^2$. I know i need to use that formula but have no idea how to ...
0
votes
0answers
28 views

Subset Sum represented as a perfect number

Can we form a set of $29$ distinct integer elements such that every subset of elements possible has a sum which is a perfect power? A perfect power is a positive integer that can be represented a p^q ...
3
votes
3answers
167 views

How should I prove $(a+b)^3= a^3+3ab(a+b)+b^3$ — Model or figure?

In what way can I prove/verify $(a+b)^3= a^3+3ab(a+b)+b^3$ ? Should I make a 3D model, or create 2D figure? In the case of 3D model, I have made $a^3$ and $b^3$; i.e cube'a' and cube'b'. I don't know ...
0
votes
1answer
22 views

Expressing units of time

How would you express 8/3 seconds as time after 3pm ? 8/3 = 2.66666 0.66*60 =40 miliseconds = 0.04 seconds so 2.04 seconds after 3 3:00:02:04 pm ? Is this correct?
1
vote
3answers
37 views

Is this triangle question missing information?

In the $\Delta KLP$, find $a+b$: My question is that: isn't some information missing from the question? Because all I can see is is that $ \usepackage{ gensymb } \angle SKP = \angle LTS = ...
0
votes
2answers
35 views

Re-writing a a differential function

I don't understand the concept of this... how do I derive a an equation written in terms of a function? How do I differentiate f(function inside) ...?
0
votes
0answers
33 views

Finding the constant of a function in terms of the gradient of a tangent.

Let $f : \Bbb R \to \Bbb R, f (x) = e^x+ k$, where $k$ is a real number. The tangent to the graph of $f$ at the point where $x = a$ passes through the point $(0, 0)$. Find the value of $k$ in terms of ...
-1
votes
2answers
32 views

How to solve $\frac12 \sec^2 \frac x2 = 1$ under restricted domain?

solve: $$\frac12 \sec^2 \left(\frac x2\right) = 1$$ and domain $x: (-\pi,\pi) \cup (\pi,3\pi)$. sec^2 (x/2) = 2 sec^2 (x/2) can be re-written as tan(x/2)^2 + 1, therefore tan^2(x/2) + 1 = 2 ...
0
votes
1answer
50 views

An algebra/linear algebra question

Suppose 8 real numbers $a,b,c,d$ and $x,y,z,w$ satisfy \begin{equation*} a^2+b^2+c^2+d^2=x^2+y^2+z^2+w^2=1,\quad ax+by+cz+dw=0. \end{equation*} Is it true that \begin{equation*} ...
0
votes
1answer
27 views

General Solution for Cosine (negative angles)

cos2(x+pi/3)=1/2 2(x+pi/3)=pi/3 x+pi/3=pi/6 x+2pi/6=pi/6 x=-pi/6 x=5pi/6 (is this step correct) ... ?? x = +/- pi/6 +kpi , k is a subset of Z x = +/- 5pi/6 +kpi , k is a subset of Z can someone ...
0
votes
1answer
34 views

Finding equations when given new center of a circle

$y = −x + \sqrt{2}$, $y = −x − \sqrt{2}$, $y = x + \sqrt{2}$, and $y = x − \sqrt{2}$. These equations determine lines, which in turn bound a diamond shaped region in the plane. Construct a diamond ...
0
votes
1answer
34 views

Gram-Schmidt method and matrices help please!

How would I use the method of Gram-Schmidt to obtain an orthonormal basis for the column space of the matrix? Any help is appreciated!
0
votes
1answer
20 views

Construct a basis from two vectorial spaces of finite dimension

Let $F$ a field, $V$ and $W$ $F$-vector spaces with dimension $n$ and $m$ respectively, $T: V \rightarrow W$ a linear transformation nonzero,then exist a basis $\beta$ and $\gamma$ of $V$ and $W$ such ...
1
vote
4answers
45 views

how to find the value of alpha and beta

here is a question : if $a$ and $b$ are the zero of polynomial $ f(x)=2x^2-9x+9$, then $1/a+1/b$ is equal to : (a) $9/2$ (b) $3/9$ (c) $1$ (d) $-1$ which is the correct answer? ...
0
votes
2answers
48 views

show that $\det(A)=0$ in this case

(a) Let $x$ and $y$ be $n\times 1$ matrices, $n \ge 1$, and let $A=xy^T$. Show that $\det(A)=0$. (b) Explain why the statment in part (a) is false if $n=1$.
2
votes
3answers
275 views

Understanding underlying algebra behind simplified expression

The solution to a linear algebra problem I'm working on reads: $$\det(A-\lambda I) = \det\begin{pmatrix}-\lambda & 1 & 0 \\ 0 & -\lambda & 1\\ 1 & -1 & ...
0
votes
1answer
81 views

Polynomial: Finding its value

If $a-b=3$, $a+b+x=2$, then find the value of $(a-b)\left(x^3-2ax^2+a^2x-(a+b)b^2\right)$ I could only substitute the value of $a-b$ there. I seriously want to try as much as I can on my own but ...
0
votes
2answers
69 views

Eigen values and Eigen vectors

Let A be a 4x4 matrix with real entries such that $ \ -1,1,2,-2 \ $ are its eigen values.If $B=A^4-5A^2+5I$ ,where $I$ denotes the 4x4 identity matrix ,then which of the following statements are ...
1
vote
0answers
32 views

Convolving two functions

I'm trying to convolve two functions $f$ and $g$. $$f(x) = e^{-\frac{{(x-p_2)}^2}{2 q_2^2}}$$ $$g(x) = \left(i_1 e^{-\frac{(a-x)^2}{2 \sigma ^2}}+j_1 e^{-\frac{(b-x)^2}{2 \sigma ^2}}\right) \left(i_0 ...
1
vote
0answers
35 views

How to approach sketching sine and cosine graphs with transformations

Any tips or suggestions in sketching these graphs quickly, and in ONE go? In exams, I don't want to spend ages re-drawing the original sine/cosine graph, one by one, following each new ...
1
vote
5answers
164 views

Algebra problem stumping me

I have recently run into an algebra problem that goes as follows. Using the digits $1$ to $9$, $$ \left\{ \begin{align} A + B + C + D &= EF \\ E + F + G + H &= CJ \\ B + G + J ...
0
votes
0answers
38 views

Simplifying a rather long expression

I'm struggling to simplify this expression: $$ i_1 j_0 \exp \left(\frac{-81 a^2-2 p_2 \left(65 a+49 b+57 p_2-114 x\right)+2 p_1 \left(16 a-65 b-49 p_2+49 x\right)+32 a b+130 a x-65 b^2+98 b x-65 ...
1
vote
2answers
33 views

A problem regarding geometric progressions

Hello my homework included this problem and I really need a hint how to solve it. It says that the numbers $a_1,a_2 \ldots a_n$ form a geometric progression. Knowing $S=a_1+a_2+\ldots+a_n$ and $P=a_1 ...
2
votes
1answer
32 views

Correct equation for this question

The question is, A small hydroelectric generating station can produce 17 MWh of energy in 12 months. AFter 4 months of operation, another generator is added. This additional generator can produce 11 ...
1
vote
3answers
77 views

Solution for $x$ with exponents?

I am trying to solve the following, $$7^{(2x+1)} + (2(3)^x) - 56 = 0$$ Should I put the 56 on the other side and get the log of both sides and is there a better way to solve this.
0
votes
1answer
38 views

a question on race, time, distance

P and Q are two points on a 1km long circular track. The distance PQ along the track is 200m. Rohan started running from P and sohan started simultaneously from Q in same direction.Both reached P ...
1
vote
1answer
24 views

Volume of parallelpiped question?

I need to find the volume of a parallelpiped. The volume is spanned by 3 vectors $$\begin{cases}a=(-5,-3,2), \\ b=(1,0,2), \\ c=(2,-1,3), \end{cases}$$ so I tried with the formula $(a \times b) \cdot ...
1
vote
1answer
64 views

Confusing rational numbers

Question: If $$x = \frac{4\sqrt{2}}{\sqrt{2}+1}$$ Then find value of, $$\frac{1}{\sqrt{2}}*(\frac{x+2}{x-2}+\frac{x+2\sqrt{2}}{x - 2\sqrt{2}})$$ My approach: I rationalized the value of $x$ to ...
0
votes
1answer
30 views

proof of this equation

How can I show the gradient of trace $(W^{T}MW)$ with respect to $W$ is equal to $MW+M^{T}W$. where W is an $m\times n$ matrix and $M$ is an $m\times m$ matrix. Can anyone help me in this case?