Bases Definition and 294 Threads

The formula my textbook provides for finding change of basis matrices is: $$b'_j = a_{1j} b_1 + \cdots + a_{nj} b_n$$ I assume, since that's the convention and also because Wikipedia itself uses this formula like this, that the first index of the c's is the row, and the second is the columns...

In the book I'm reading, Before Machine Learning, by Jorge Brasil, I'm on the section that introduces bases for vector spaces. The author gives the example of a vector space with two vectors ##\vec i## and ##\vec j## forming the basis where ##\vec i = (1,0)## and ##\vec j = (0,1)## He then says...

The correct answer to this problem is: ##\sigma = \varepsilon_0E\frac{\varepsilon-1}{\varepsilon}## Here is my attempt to solve it, please tell me what is my mistake? ##E_{in} = E_{out} - E_{ind}## ##E_{ind} = E_{out} - E_{in}## ##E_{in} = \frac{E_{out}}{\varepsilon}## ##E_{ind} = E_{out} -...

Let ##H## be a Hilbert space with an orthonormal basis ##\{x_n\}_{n\in \mathbb{N}}##. Suppose ##\{y_n\}_{n\in \mathbb{N}}## is an orthonormal set in ##H## such that $$\sum_{n = 1}^\infty \|x_n - y_n\|^2 < \infty$$ Show that ##\{y_n\}_{n\in \mathbb{N}}## must also be an orthonormal basis.

Hi, I tutor maths to High School students. I had a question today that I was unsure of. Can the natural log be to the base 2? The student brought the question to me from their maths exam where the question was: Differentiate ln(base2) x^2 If the natural log is the inverse of e then how does...

Hello Forum, I am trying to get a grasp of the topic (new to me) of dictionary and dictionary learning. In general, we express a vector ##A## using a basis. A basis is a complete set of vectors that we can use to expand any other vector as a linear combination of the basis vectors. For example...

$\tiny{KAM}$ $\log_9{(x+1)}+3\log_3{x}=14$ ok not sure as to best approach to this assume change the base 9?

Hey! :giggle: Let $1\leq n\in \mathbb{N}$ and $(p_0,\ldots , p_n)$ an affine basis (that means that the vectors $p_1-p_0, p_2-p_0,\ldots ,p_n-p_0$ build a basis of $\mathbb{R}^n$. (a) Give a geometric description of affine bases of $\mathbb{R}^n$ for $1\leq n\leq 3$. (b) For all $v\in...

I have been wondering lately about the survivability of large bases on airless moons or planets. I can’t think of any way to protect a base on the moon or Mars from a deranged individual determined to kill everybody there. It seems to me that there will be many occupations on a base like that...

I am not completely sure what the formulas ##v_j = v^a\frac {\partial x^j} {\partial \chi^a}## and ##v^b = v^a\frac {\partial \chi^b} {\partial x^j}## mean. Is ##v_j## the j:th cartesian component of the vector ##\vec v## or could it hold for other bases as well? What does the second equation...

Consider two entangled spin half particles given by the generic form of Bell Equation in Z-axis: ##\psi = (a\uparrow \uparrow + b\downarrow \downarrow)## where ##a^2+b^2=1## In a (2D) planer rotated (by an angle ##\theta##) direction the new equation can be given by: ##|\psi \rangle =...

Hey! 😊 We have tha matrix \begin{equation*}A:=\begin{pmatrix}1 & 5 & 8 & 2 \\ 2 & 4 & 6 & 0 \\ 3 & 3 & 8 & 2 \\ 4 & 2 & 6 & 0 \\ 5 & 1 & 8 & 2\end{pmatrix}\in \mathbb{R}^{5\times 4}\end{equation*} Determine a basis $B$ of $\mathbb{R}^4$ and a basis $C$ of $\mathbb{R}^5$ such that...

Summary:: finding ml of two solutions by the final pH i have a NaOAc 0.1M and HOAc 0.1M , together the volume of the solutions is 20ml and the pH is 4. I need to find the volume of each solution. I've tried to solve it for hours with no successes. i found the H+ concentration (-log(h)=4 ), it...

This question says: An HA weak acid solution with a molarity of 0.1M is dissolved in water. In the new solution, is the molarity of OH- greater than the H3O+ molarity, or the opposite? Or are they equal? I came up with two possible answers: 1. [H3O+]>[OH-] because there are no hydroxides...

The question says: A solution of highly acidic HA is given, with a molarity of 1M. Is it true that [A-]>[H3O+] or not? I simply don't understand why the hydronium is mentioned and i don't know how to find the molarity of these two individually.

For equilibrium, using ##\Sigma \vec F = 0##, we get ##n_1 + n_2 = 300\; \text{N}##. Taking the system as a whole and applying ##\Sigma \vec \tau = 0## about the hinge (pin) at the top from where the load is hung, we get ##n_1 \times (0.8) \times 4 = n_2 \times (0.6) \times 3##, by taking...

I am reading Andrew McInerney's book: First Steps in Differential Geometry: Riemannian, Contact, Symplectic ... and I am focused on Chapter 3: Advanced Calculus ... and in particular on Section 3.3: Geometric Sets and Subspaces of ##T_p ( \mathbb{R}^n )## ... ... I need help with...

I'm doing problems on finding row and column spaces. My textbook tells me to find the echelon form of the matrix, and then to identify the bases. My question is, can I reduce the matrix to reduced echelon form to get the bases? I have the same question about bases for the solution space.

Homework Statement On ##L_2[0,2\pi]## where ##e = \{ 1/\sqrt{2 \pi},1/\sqrt{\pi}\sin x,1/\sqrt{2 \pi}\cos x \}##. Given ##f(x) = x##, find ##Pr_e f##. Homework Equations See solution. The Attempt at a Solution I take $$e \cdot \int_0^{2\pi} e f(x) \, dx = \pi - 2 \sin x.$$ Look correct?

Hi, could you please explain me, how to copmute the yellow parts? Thank you very much.

Homework Statement [/B] (Working through a problem from a practice set for which I have a solution available, but still don't understand. I get the same answer as they do for part a, but get lost in part b, I think. Relevant portions below) Consider a two-state quantum system. In the...

In the exercises on differential forms I often find expressions such as $$ \omega = 3xz\;dx - 7y^2z\;dy + 2x^2y\;dz $$ but this is only correct if we're in "flat" space, right? In general, a differential ##1##-form associates a covector with each point of ##M##. If we use some coordinates...

The log to the base 10 of 1000000 is the number 6. this is a much contracted number in terms of length. But the log to the base 10 of 1234567 is 6.0915146640862625...this is an even longer set of digits than the first example , despite the two original numbers both starting with the same length...

Hello. I am wondering how I can find the area of a trapezoid from its two legs and bases. My problem: ABCD is a trapezium with AB parallel to CD such that AB = 5, BC = 3, CD = 10 and AD = 4. What is the area of ABCD? If we trace a straight line from A down parallel to the height of the...

Homework Statement I don't want to clog up the forums with a few "small" problems so I am lumping them together here. 2. Let ##T:P^1 → R \text { be given by } T(p(x)) = \int^b_a p(x)dx##. Describe Ker(T) using set notation. 3. Let ##H = \left\{f ∈ C[a, b] | f'(x) ≥ 0 ~\text for...

Homework Statement Please see attached file. I'm not quite sure if I'm on the right track here. I think the basis for F is throwing me off as well as T(f). Please advise. Thanks! Homework EquationsThe Attempt at a Solution

Homework Statement Let ##W_1=\langle (1,2,3,6),(4,-1,3,6)(5,1,6,12))\rangle## and ##W_2=\langle (1,-1,1,1),(2,-1,4,5)\rangle## be subspaces of ##\Bbb{R}^4##. Find the bases for ##W_1\cap W_2## and ##W_1+W_2##. Homework Equations...

Homework Statement So I have these two Matrices: M = \begin{pmatrix} a & -a-b \\ 0 & a \\ \end{pmatrix} and N = \begin{pmatrix} c & 0 \\ d & -c \\ \end{pmatrix} Where a,b,c,d ∈ ℝ Find a base for M, N, M +N and M ∩ N. Homework Equations I know the 8 axioms about the vector spaces. The...

Hello In our Quantum Mechanics lecture we have been discussing a simplified model of the Stern-Gerlach experiment. Let ##|+>## and ##|->## denote an electron that is "spin up" and "spin down" (with respect to ##\hat{z}##), respectively. Our professor then asserted that ##|+>## and ##|->## acted...

Homework Statement It's a synthesis, dedicated more to carbonyl reactions. However, there is this step where you deduce the structure of an elongated terminal alkyne and then they tell you to add 1. MeMgBr 2. Formaldehyde. Homework Equations Usually organometallics are used to transform the...

I am trying to figure how one arrives at the following: dxμ∂ν = ∂xμ/∂xν = δμν Where, dxμ is the gradient of the coordinate functions = basis of cotangent space ∂ν = basis of tangent space I know that dual vectors 'eat' vectors to produce scalars. Is this demonstrated by absorbing d into ∂...

Hey! :o We have the subset $X_i$ of $\mathbb{R}^2$: $$X_1 := \{(x,y) \in \mathbb{R}^2 : x + y = 0\}; \\ X_2 := \{(x,y) \in \mathbb{R}^2 : x + y = 1\} \\ X3 := \{(x,y) \in\mathbb{R}^2 : x^2 + y^2 = 0\}; \\ X4 := \{(x,y) \in \mathbb{R}^2 : x^2- y^2 = 0\}$$ We want to check which of these sets...

Hello! I am a bit confused about non-coordinate basis. I understand the way they are defined (I think) and the main purpose is to get on a manifold a coordinate system that is orthonormal at any point on the manifold (right?). So if you have a coordinate basis ##e_\alpha##, you get to a...

Hello, In the case of 2D vector spaces, every vector member of the vector space can be expressed as a linear combination of two independent vectors which together form a basis. There are infinitely many possible and valid bases, each containing two independent vectors (not necessarily...

Going through a problem and and I keep getting it wrong and I'm not sure why. In a part of the problem, the expression ##\left(-3\right)\left(-r^4\right)\left(-s^5\right)## comes up and the solution that it's giving me is ##-3r^4s^5## Wouldn't the last factor be ##-s^5## since the power of a...

Homework Statement Linear Algebra Problem: Solving for Euler between two ordered bases I've got a problem I need to solve, but I can't find a clean solution. Let me see if I can outline the problem somewhat clearly. Okay, all of this will be in 3D space. In this space, we can define some...

Hello! Why do we need to impose a change on the basis vector, when going from a reference frame to another. I understand that the components of the vector and the basis change using inverse matrices (the components use a matrix and the vector basis the inverse). But the transformation condition...

I just read something that I do not want to misinterpret. If there are two orthonormal basis that span the same space, which I think implies that each basis can be written in terms of the other basis, then measurements made with respect to each basis will not commute? Does this mean that...

Hi folks, When you study Mendelian laws you learn about dominant alleles of a gene, A, or recessive alleles of a gene. My question is, What is its relation to DNA? -So in terms of bases we have A, T, C and G. And I know a gene may be in a chromosome and it may contain several million of...

Homework Statement I'm trying to understand negative bases raised to rational powers, when calculating principle roots for real numbers. I'm not worried about complex solutions numbers at this stage. I just can't find a concise explanation I can understand anywhere. I'm self learning as an...

All natural life uses the same four bases in its DNA: A paired with T and C paired with G. Scientists worked on adding more bases. Just putting them into DNA is not hard, the challenging part is to keep them there: They should not get removed/replaced during reproduction. This has now been...

Which of the following statements about weak acids and weak bases are false? (i) The strength of a weak acid decreases as itspKa decreases. (ii) Weak bases react with water to produce a small amount of a strong base. (iii) The strength of a base decreases as the strength of its conjugate acid...

Hello. This is how every number in the decimal system is expressed: I had understood this topic earlier but as I was revising it today I have become confused somewhat. I know that for the decimal system, we have 9 digits. I understand this: - When we use a base between 1-10, we do not...

What are the bases for the adjoint representation for SU(3)?

According to this video the length of basis is r. It grows as we further from the origin . Why?

Hi, I have a doubt on the topic of bases and alkalis. I have learned that a alkali is a soluble base so does that mean Sodium Oxide(Solid) is a alkali and Lithium Hydroxide(Solid) is a alkali. Or are they considered alkali when they are dissolved? For example, Sodium Oxide becomes sodium...

I am reading Paul E. Bland's book "Rings and Their Modules ... Currently I am focused on Section 2.2 Free Modules ... ... I need some help in order to fully understand Bland's Example on page 56 concerning directly finite and directly infinite R-modules ... ... Bland's Example on page 56...

I am reading Paul E. Bland's book "Rings and Their Modules ... Currently I am focused on Section 2.2 Free Modules ... ... I need some help in order to fully understand the proof of the equivalence of (1) and (3) in Proposition 2.2.3 ... Proposition 2.2.3 and its proof reads as follows: Bland...

I am reading Paul E. Bland's book "Rings and Their Modules ... Currently I am focused on Section 2.2 Free Modules ... ... I need some help in order to fully understand the proof of the equivalence of (1) and (3) in Proposition 2.2.3 ... Proposition 2.2.3 and its proof reads as follows:Bland...

Homework Statement Write an expression containing a single radical and simplify. Homework Equations \sqrt[4]{xy}\sqrt[3]{x^2{y}} The Attempt at a Solution I can't add the exponents and I can't multiply the bases. I can't take anything out of the radicals to make the bases the same. I have no...