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I was thinking of functions who have differential equations in which the function is 'factorable' from first order differential equation. Eg: Functions satisfying the equation:
$\frac{d y}{d x} = η (x) y (x)$
Example functions:
$y = e^{a x}, y = \frac{1}{1 - x}$ .. etc
Now, I notice that the solution to the differential equation is:
$y = e^{\int η d x}$
But here is the interesting thing, the above expression has the same form of the function which is used when we solve linear differential equation

Cian Orr 2022-02-24

Theorem:
Given a system of linear equations $A x = b$ where
$A \in M_{m \times n} (R), x \in R_{col}^{n}, b \in R_{col}^{m}$ .
Deduce that a solution x exists if and only if $rank (A ∣ b) = rank (A)$ where $A ∣ b$ is the augmented coefficient matrix of this system
I am having trouble proving the above theorem from my Linear Algebra course, I understand that A|b must reduce under elementary row operations to a form which is consistent but I don't understand exactly why the matrix A|b need have the same rank as A for this to happen.
Please correct me if I am mistaken

Asiya Holder 2022-02-24

"Convert the following system of linear equations into an augmented matrix and then solve it"
$x_{1} + x_{2} - x_{3} - x_{4} = 7$
$2 x_{1} - x_{2} + 3 x_{3} + x_{4} = 1$
$x_{1} - 5 x_{2} + 9 x_{3} - x_{4} = 3$
My approach: So the augmented matrix, A, is
$a u g A = [\begin{matrix} 1 & 1 & - 1 & - 1 & 7 \\ 2 & - 1 & 3 & 1 & 1 \\ 1 & - 5 & 9 & - 1 & 3 \end{matrix}]$
Then we perform elementary row operations to reduce the matrix into row echelon form:
$B = [\begin{matrix} 1 & 1 & - 1 & - 1 & 7 \\ 0 & - 3 & 5 & 3 & - 13 \\ 0 & - 6 & 10 & 0 & - 4 \end{matrix}]$
$C = [\begin{matrix} 1 & 1 & - 1 & - 1 & 7 \\ 0 & - 3 & 5 & 3 & - 13 \\ 0 & 0 & 0 & - 6 & 22 \end{matrix}]$
Matrix C is our system of linear equations in echelon form. From the last row of matrix C,
$- 6 x_{4} = 22$
$x_{4} = \frac{- 11}{3}$
Now we let $x_{3} = t$ for t is any real number
Then we solve the two equations from the first two rows of matrix C,
$x_{2} = \frac{1}{3} (- 2 - 5 t)$
$x_{1} = \frac{8}{3} t + 2 \frac{2}{3}$
SO this should be the solutions to the equations. However, according to the answer key the solution is the empty set. I inputted the regular matrix into a determinant calculator and indeed the determinant was zero. However, I managed to get a solution to the system of linear equations. Can someone tell me where I went wrong?

iristh3virusoo2 2022-02-24

I tried researching and found that I can use a system of linear equations and solve by an inverse matrix to find the cubic equation given 4 points which satisfy the function f(x) of the general form $f (x) = a x^{3} + b x^{2} + c x + d$
I can also find a cubic of the form $a x^{3} + d$ with no $x^{2}$ or x term from 2 points, however I was wondering how one would go about finding a full general form cubic given only the minimum and maximum.
Example minimums could be (-1,4) and (2,3)

Theresa Chung 2022-02-24

I know how to use methods like system of linear equations to show that any vector in $R^{n}$ can be expressed in the form of unique linear combinations of a given set of n linearly independent vectors, but how to expand this result to that any given set of exactly n linearly independent vectors must form a basis of $R^{n}$ space? How to understand it conceptually?
I saw there are similar questions, but the answers in those posts do not really convince me.
Thanks in advance.

Erik Sears 2022-02-24

Let X be a smooth hypersurface in Projective space $P^{n}$ of degree d defined by the equation f=0. Given that we have a vector bundle E of rank $r \geq 1$ on X such that we have the following exact sequence on $P^{n}$ :
$0 \to O {(- 1)}^{r d} \to O^{r d} \to E \to 0$ .
My question is as follows. What is the morphism from $O {(- 1)}^{r d} \to O^{r d}$ ? A paper indicated that it is given by a $r d \times r d$ matrix of linear forms. Why is this? I am not able to see it. If this is so, can we say where in $P^{n}$ the determinant of that matrix vanishes?

Odompombagnom6ng 2022-02-24

The question: 3 solutions of a certain 2nd order non homogenous linear equation are:
$ψ_{1} (t) = t^{2}$
$ψ_{2} (t) = t^{2} + e^{2 t}$
$ψ_{3} (t) = 1 + t^{2} + 2 e^{2 t}$
Find the general solution of the equation.
My attempt: (I closely followed an example from the book)
so $ψ_{2} (t) - ψ_{1} (t) = e^{2 t}$ and $ψ_{3} (t) - ψ_{2} (t) = 1 + e^{2 t}$ are solutions of the corresponding homogenous equation. Next I need to show that these are linearly independent in order to use the theorem to find the general solution. I assume they are, but I am not sure how to exactly show that?
And then using that theorem: Every solution is in the form of $y (t) = c_{1} y_{1} (t) + c_{2} y_{2} (t) + ψ (t)$ so
$y (t) = c_{1} y_{1} (t) + c_{2} y_{2} (t) + ψ (t)$
$y (t) = c_{1} e^{2 t} + c_{2} (1 + e^{2 t}) + t^{2}$
$y (t) = (c_{1} + c_{2}) e^{2 t} + c_{2} + t^{2}$
I am not really sure about my answer, particularly during the part of assuming linear independence.

Adrienne Rowe 2022-02-24

The latter form 'can also be related to the two-point form, where $T = p - h, U = h, V = q - k$ and $W = k$ . How would one derive this relation — i. e.: how would one derive the parametric form of the equation for a line from the two-point form of the equation for a line?

Tashan Flores 2022-02-24

I have no idea how to solve it. Should be linear equation of order one since I am passing through this chapter, but I can't put into the form of
$y^{'} + P (x) y = Q (x)$
Here is the equation:
$(2 x y + x^{2} + x^{4}) d x - (1 + x^{2}) d y = 0$
It is not exact since partial derivatives are not equal.
Any help would be appreciated.

Huzaifah Hays 2022-02-24

Establish the conditions under which the equations
$a x + b y + c z = q - r; b x + c y + a z = r - p; c x + a y + b z = p - q$ ,
are consistent. I am aware that by taking the system to echelon can get me the rank of the coefficient matrix and the augmented matrix. But that turns out to be a very cumbersome process and the terms in the echelon form are very big. Is there other ways to establish the conditions for consistency which are less cumbersome and time consuming?

VorkDesegroorrta 2022-02-24

I have a linear problem with double derivate of space and time, which has Dirichlet boundary condition in ${(1)}_{2}$ and Neumann's boundary condition in ${(1)}_{3}$ :
$\frac{δ^{2} u}{δ t^{2}} - c^{2} Δ u = f in Ω$
$u = 0 on δ Ω or$
$\nabla u \cdot n = 0 on δ Ω$
with conditions
$u (0, x) = u_{0} (x) in Ω$
$\frac{δ u}{δ t} (0, x) = v_{0} (x) in Ω$
Which is the weak form of the problem?
My attempt:
$\int (\frac{δ^{2} u}{δ t^{2}} - c^{2} Δ u) \cdot v d x = \int f v \cdot d x$ ,
for $\forall v \in V_{n}$ . Handling two different parts separately
$\int_{Ω} \frac{δ^{2} u}{δ t^{2}} v d V + c^{2} [\int_{Ω} \nabla u \cdot \nabla v d V - \int_{δ Ω} \nabla u \cdot \hat{n} \cdot v d A]$
$= \int f v \cdot d x$

Nyah Conrad 2022-02-24

If I get the slope of a line, and one point that is on it, then, are they define exactly ONE line?
The point–slope form of linear equations $(y - y_{1} = m (x - x_{1}))$ need only the slope and the coordinates of that given point. And this equation describes exactly one line.
Is it enough?

Layla-Rose Ellison 2022-02-24

Why is the equation $\frac{d y}{d x} + P (x) y = Q (x)$ said to be standard form?
Well, I know that in linear differential equation the variable and its derivatives are raised to power of 1 or 0. But I am confused where did the standard form of linear differential equation came form?
That is, why is the equation $\frac{d y}{d x} + P (x) y = Q (x)$ said to be standard form?

demonlikw4 2022-02-24

There I have found from an old tutorial notes from a Linear Algebra course I am taking that I am finding the question rather cumbersome. I haven't come across any elementary matrices problem set in this way before. I have tried to using Gauss-Jordan Elimination to reduce matrix A, whilst performing the row operations on the corresponding elementary matrix. Although, I am unsure if this is the correct approach as the problem notes that RREF calculation is not necessary.
If [A|b] denotes the augmented matrix which corresponds to the system of linear equations
$w - y = - 2$
$- w + x + z = 5$
$2 x - y + 4 z = 1$ ,
determine the $3 \times 3$ matrix E such that [EA|Eb] is the reduced row echelon form of [A|b]. Note: You are asked to calculate the matrix E but you are not asked to calculate the reduced row echelon form.
Any help on how to approach this question or determine the matrix E would be highly appreciated.

Myah Fuller 2022-02-23

Suppose I have the linear equation
$a x + b y = c (1)$
where $(a, b, c) \in R^{3}$ . Let $W = {a, b, c) \in R^{3} : (1) is consistent}$ .
Is $W$ a subspace of $R^{3}$ ?
Apparently the answer is no, but I am unable to find a counter-example. If anything, my reasoning (below) tells me W is a subspace.
1. $(0, 0, 0) \in W$ .
2.If $(a, b, c) \in W$ , then $(α a, α b, α c) \in W$ for any non-zero $α \in R$
3. If $(a, b, c), (a^{'}, b^{'}, c^{'}) \in W$ , then
$(a + a^{'}, b + b^{'}, c + c^{'}) \in W$

Arnold Herring 2022-02-23

Prove that two homogeneous systems of linear equations are equivalent iff they have the same solution set.
My definition of equivalent of linear equations is that each equation of system A is a linear combination of the equations of system B and converse.
So i have a bit trouble proving this statement in general as $m \times n$ form. I worked with their coefficient matrices and if i show that these two are row equivalent the problem is solved. Is there any method to use for this? If not what's the general prove.

klancimarsn 2022-02-23

a) Determine a condition under which (x, y, z) is a linear combination of [-3, 5, -3], [-9, 11, -3], [-6, 8, -3]? Your condition should take the shape of a linear equation. I have the theorem that every vector (x, y, z) in $R^{3}$ is a linear combination of $x (1, 0, 0) + y (0, 1, 0) + z (0, 0, 1)$ .

Ravi Stein 2022-02-23

Let A be an abelian group, $a_{i}$ variables indexed with some arbitrary set I and assume we have an infinite set E of linear equations in finitely many variables of the form
$n_{1} a_{1} + \dots + n_{k} a_{k} = b$
with $n_{i} \in Z, b \in A$ .
If this system has no solution, then is there already a finite subset of equations in E which are inconsistent? So if any finite subset of equations is solvable, is E solvable?
This is reminiscent of the compactness theorem in 1st order logic, but of course you can't apply this for a concrete group. Am I missing something? I assume that this has an easy proof or a counterexample. What happens if we presume particularly nice groups (divisible, free) or vector spaces? Then this question is just about linear systems of equations.

Miruna Atherton 2022-02-23

If the system of linear equations
$x + k y + 3 z = 0$
$3 x + k y - 2 z = 0$
$2 x + 4 y - 3 z = 0$
has a non-zero solution $(x, y, z)$ , then $\frac{x z}{y^{2}}$ is equal to:
$(1) 30$
$(2) - 30$
$(3) - 10$
$(4) 10$
In the solution given, the determinant of matrix formed from the coefficients of x,y,z in the three equations is equated to 0 and the corresponding value for k is found.
However, as far as I know the condition for non-zero solutions is that the determinant must not equal 0. Is there anything I am missing? I am not asking for the solution. I only need some clarity in the concept. Any help is greatly appreciated.

iristh3virusoo2 2022-02-23

I am trying to convert part of an equation from its log form into a linear form. Specifically, I am trying to convert $10^{4 \log (x)}$ , into $x^{4}$ , but I'm really unsure of how to get from this first stage to the second. My experience with logarithms and exponents is limited, though I believe that $10^{4 \log (x)}$ can be re-written as $10^{4} + 10^{\log (x)}$ , but I'm not sure that this helps my plight! Any very basic guidance would be greatly appreciated.

When you are a high school student or someone already enrolled in college, even those simple research projects will include Algebra 1 equations, which means that getting some help cannot be avoided. Thankfully, you can access Algebra 1 problems and answers free of charge, which will let you see what Algebra 1 practice problems can be used for certain questions. Just remember to check calculations twice as these can easily get complex. The majority of Algebra 1 answers will be familiar to most students dealing with financial concepts as it is not related to advanced Math problems in most scenarios.

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