What is Bases: Definition and 293 Discussions

The Business Association of Stanford Entrepreneurial Students (BASES) is a student group at Stanford University focusing on business and entrepreneurial activities. One of the largest student-run entrepreneurship organizations in the world, BASES' mission is to promote entrepreneurship education at Stanford University and to empower student entrepreneurs by bringing together the worlds of entrepreneurship, academia, and industry. BASES organizes the flagship 150K Challenge, Entrepreneurial Thought Leaders' Seminar, the SVI Hackspace, E-Bootcamp, and the Freshman Battalion.BASES was founded in 1996 by a group of five Stanford engineers. The organization works in partnership with Silicon Valley's venture capitalists and law firms to provide a variety of entrepreneurial services to Stanford students.

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  1. N

    B Understanding Bases of a Vector Space

    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...
  2. rokiboxofficial Ref

    Surface density of the charges induced on the bases of the cylinder

    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} -...
  3. Euge

    POTW Orthonormal Bases on Hilbert Spaces

    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.
  4. YouAreAwesome

    B Can a log have multiple bases?

    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...
  5. F

    I What is the difference between a complete basis and an overcomplete dictionary?

    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...
  6. karush

    MHB What is the solution to the logarithmic equation with different bases?

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

    MHB Exploring Affine Bases in $\mathbb{R}^n$

    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...
  8. Whipley Snidelash

    Surviving Large Bases on Airless Worlds: Is It Possible?

    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...
  9. K

    Conversion between vector components in different coordinate systems

    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...
  10. R

    I Measurement of an entangled Particles in two Different Bases

    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 =...
  11. M

    MHB How to determine the bases B and C

    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...
  12. sergey_le

    Calculations involving acids and bases

    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...
  13. H

    Comparing a hydronium with a hydroxide in a weak acid solution (0.1M)

    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...
  14. H

    Is [A-] greater than [H3O+] in a 1M solution of highly acidic HA?

    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.
  15. brotherbobby

    Normal reactions at the bases of two light supports

    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...
  16. Math Amateur

    I Bases for Tangent Spaces and Subspaces - McInerney Theorem 3.3.14

    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...
  17. A

    B Finding Bases for Row and Column Spaces

    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.
  18. M

    Projections of functions and bases

    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?
  19. Teri

    I Hyperfine structure - Hamiltonian and bases

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

    Quantum - Two State Problem in different bases

    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...
  21. K

    I Understanding Differential Forms and Basis Vectors in Curved Space

    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...
  22. M

    MHB Using Nested Logs with differing bases to contract a number

    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...
  23. ISamson

    Finding height and area of trapezoid from its legs and bases

    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...
  24. bornofflame

    [LinAlg] Set Notation, Subspaces, Bases, Ranges, & Kernels

    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...
  25. R

    Matrix representation relative to bases

    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
  26. T

    Finding bases for ##W_1\cap W_2## and ##W_1+W_2##

    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...
  27. Alex Langevub

    Is the zero Matrix a vector space?

    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...
  28. M

    I Understanding Spin States in Hilbert Space

    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...
  29. Upupumiau

    Can organometallic act as bases?

    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...
  30. N

    A How do dual vectors and tangent bases relate in coordinate functions?

    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 ∂...
  31. M

    MHB Describe bases for the span of sets

    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...
  32. S

    I Non-Coordinate Bases Explained

    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...
  33. F

    I Bases, operators and eigenvectors

    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...
  34. D

    B Understanding Negative Exponents and Bases in Algebraic Expressions

    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...
  35. Elroy

    Linear Algebra Problem: Solving for Euler between two ordered bases

    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...
  36. S

    I Transform Bases for 4-Vectors in Ref. Frames

    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...
  37. M

    I Two orthonormal bases that span the same space

    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...
  38. J

    Relation between Mendelian genes and bases?

    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...
  39. S

    When are Negative Bases Raised to Rational Powers Undefined?

    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...
  40. mfb

    A, T, C, G: Add X and Y (DNA bases)

    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...
  41. K

    Weak bases react with water to produce strong base?

    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...
  42. L

    I Some questions about bases and the decimal system.

    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...
  43. N

    I Bases for SU(3) Adjoint representation

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

    I Length of bases in Polar coordinates

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

    Alkalis and bases differentiation

    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...
  46. Math Amateur

    MHB Understanding Bland's Example: Free Modules & Directly Finite/Infinite R-Modules

    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...
  47. Math Amateur

    I Free Modules, Bases and Direct Sums/Products

    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...
  48. Math Amateur

    MHB Free Modules, Bases and Direct Sums/Products - Bland, Proposition 2.2.3

    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...
  49. L

    Multiplying different bases with different exponents

    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...
  50. RJLiberator

    PDE: Proving that a set is an orthogonal bases for L2

    Homework Statement Show that the set {sin(nx)} from n=1 to n=∞ is orthogonal bases for L^2(0, π). Homework EquationsThe Attempt at a Solution Proof: Let f(x)= sin(nx), consider scalar product in L^2(0, π) (ƒ_n , ƒ_m) = \int_{0}^π ƒ_n (x) ƒ_m (x) \, dx = \int_{0}^π sin(nx)sin(mx) \, dx =...