# 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.

View More On Wikipedia.org
1. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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?

32. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### 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. ### I Bases for SU(3) Adjoint representation

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