Bases Definition and 294 Threads

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. M

    Question relating to homogeneous system, subspaces and bases.

    Homework Statement Let A be an m x n matrix such that the homogeneous system Ax=0 has only the trivial solution. a. Does it follow that every system Ax=b is consistent? b. Does it follow that every consistent system Ax=b has a unique solution? The Attempt at a Solution So if the homogeneous...
  2. R

    Physics bases future technologies

    I have recently read some articles that deal with future technologies such as invisibility and teleportation. There are some magazines, websites and scientists such as Micho Kaku who make future predictions based on the latest cutting-edge technologies and scientific data. My question is...
  3. U

    Understanding Lewis Acids & Bases: HCl, H2SO4, H3PO4, CO2, HCN

    hi all! I am not able to understand what's a Lewis acid and a Lewis base. tell me if these are Lewis acids or Lewis bases and tell me the reason. HCl, H2SO4, H3PO4, CO2, HCN. my approach: first 3 I'm not able to decide. 4th one I thought to be Lewis base as 2lone pairs of electrons left in...
  4. D

    Tensor Analysis: quadric, bases, MOI, bisectors

    Homework Statement 1) Let e, e' be two orthonormal bases(not necessarily the standard basis). Prove that the matrix S = [Id]e-e' (direct transformation from e to e') is orthogonal, i.e. SS^T = S^TS = [Id]_3. SS^T is S * S transpose. 2) The moment of inertia tensor of a right circular cylinder...
  5. B

    How Do Acids and Bases Interact in Reactions?

    Sorry guys, I can't seem to grasp the concept: It deals with acids and bases. When the acid donates the proton, to the base, how do we know how many is transferred and how many the base can accept? This is where I am stumped. Homework Statement What ions are formed when nitric...
  6. Rasalhague

    Garrity's Formula: Ordering Covariant Alternating Tensors

    I have a question about Garrity's formula at the top of p. 125, here, for a function from the set of 2-form fields to the set of tangent vector fields, together with the formula on p. 123 for the exterior derivative of a 1-form field and Theorem 6.3.1 on p. 125 (Garrity: All the Mathematics you...
  7. F

    Finding matrix relative to different bases

    Homework Statement Let B = {b1, b2, b3} be a basis for a vector space V and T : V -> R2 be a linear transformation with the property that T(x1b1 + x2b2 + x3b3) = 2x1 - 4x2 + 5x3 0x1 - 1x2 + 3x3 Find the matrix for T relative to B and the standard basis for R2. Homework...
  8. C

    MIPS Help (converting numbers to diff bases)

    Alright, I have a program that converts numbers from base 2-10 and it works just fine but I can not get the program to convert past base 10 whenever i input any char such as 1A it says invalid input. I have tried changing some of the lines in the code but it ends with errors. Right now it is...
  9. S

    Representing a wavefunction using bases

    Can someone please explain why the representation of a wavefunction as an expansion of basis eigenfunctions actually gives us something of physical meaning? For example, it can tell us the probabilities of measuring a particular eigenvalue (depending on the expansion coefficients)... I mean its...
  10. I

    Qustion about leaving groups and bases

    If I had a nucleophile, say the conjugate base of benzene, and a molecule with a bromide, and a primary alcohol. Would the nucleophile attack the carbon bound to the bromide, or the proton on the alcohol? If it's hard to follow I can try to make a picture. Basically I want to add Isobutanol...
  11. K

    Right translations of orthonormal bases.

    Homework Statement Let G be a Lie group and L be its Lie algebra. Further assume that TG is equiped with a metric tensor g; that is, a (0,2)-covariant tensor that is symmetric and positive-definite on fibres of TG. Assume that \{ H_i\}_{i=1}^m are an orthonormal basis for for L. If R_X: G...
  12. Z

    Find bases for the following subspace of F^5

    Homework Statement Find bases for the following subspaces of F^5: W1 = {(a1, a2, a3, a4, a5) E F^5 : a1 - a3 - a4 = 0} and W2 = {(a1, a2, a3, a4, a5) E F^5: a2 = a3 = a4 and a1 + a5 = 0} 2. The attempt at a solution Well, I understand a basis is the maximum amount of vectors...
  13. D

    Understanding Bases in Topology: Examining Subsets of Real Numbers

    I have a Topology midterm tomorrow and I'm going through exercises in my book. Perhaps someone could let me know whether I ought to make a thread for each question or if I may continue adding to this thread... Determine which of the following collections of subsets of R are bases: a.) C1 =...
  14. C

    Are Amines Considered Strong or Weak Acids and Bases in Mastering Chemistry?

    So, it's for Mastering Chemistry and I need to sort out the strong acids weak acids strong bases and weak bases. It says that two are wrong though I've looked them all up and don't know which ones are wrong. [PLAIN]http://bit.ly/fREfFS
  15. J

    Quick question about matrices & bases?

    Say I was given a 2x2 matrix made from a certain basis {|x\rangle, |y\rangle} , and I split that matrix into two parts, one being the diagonal part and one being the off-diagonal part. for example, if I had H = H_0 + W = \left(\begin{array}{cc}a&c\\b&d\end{array}\right) =...
  16. M

    Orthonormal Bases: Determining Coefficients for Arbitrary Vector

    If we have a vector that can be expressed in terms of some finite list of basis elements. If we have an orthonormal basis for a vector space V, then a vector v can be expressed as <v,e1>e1 +...+ <v,en>en. This appears to be widely used for many results (such as Gram-Schmidt), but the...
  17. A

    What Is the Basis for the Kernel of These Linear Transformations?

    Find bases for the kernal of the following: T(x1,x2,x3,x4) = (x1+x2+x3,-x3,x1+x2) Any help would be great thank you
  18. P

    Linearly Independent Sets and Bases

    Homework Statement V is a subspace of Rn and S={v1,...,vk} is a set of linearly independent vector in V. I have to prove that any list of linearly independent vectors can be extended to a basis for V. Homework Equations None that I can think of. The Attempt at a Solution So to be...
  19. S

    What is the best problem to solve for an IB Design Lab on acids and bases?

    I need some help with developing an idea for my design lab. it is a lab where we must come up iwth a problem to solve on our own. This time we are to discover something new about acids and bases. What would be a good problem? thank you!
  20. P

    How do I classify acids and bases in chemistry?

    Can anyone explain to me how to determine whether a compound is a strong acid, weak acid, strong base, weak base, or amphoteric? Thanks!
  21. S

    Difference between the 2 bases pairs

    hi, i wonder if there is any clear chemical difference between the bases pairs CG and AT? is one more reactive than the other one for example? is there any chemical difference that is used? i would be very grateful for any reply! thanks!
  22. T

    Find Basis for Uperp from x1, x2, x3: Orthogonal Bases

    Homework Statement Let U be a subspace of R4 and let S={x1, x2, x3} be an orthogonal basis of U. Given x1, x2, x3, find a basis for Uperp (the subspace containing all vectors orthogonal to all vectors in U). I am actually given three vectors x1, x2, x3, but I am looking more to...
  23. J

    Galois Fields, Grobner Bases, N queens

    Hi all. Thank you for taking the time to read this. I am currently studying Grobner Bases and I've been given a problem that I'm struggling to find any resources for. I am interested in find if a certain n for the n queens problem have solutions. Now there are SAT solvers and such which can...
  24. Q

    Understanding Pauli X Matrix in Z & X Bases

    hi, anyone can enlighten me why the pauli x matrix in the z basis is given as 0 1 1 0 while it in the x basis, it is given as 1 0 0 -1 is there a formula or something to calculate this? and how does one know that a matrix is in the z basis or the x basis? thanks a lot!
  25. M

    Is the Change of Basis Matrix in My Book Wrong?

    Homework Statement I have posted this problem on another website (mathhelpforum) but have received no replies. I don't know whether this is because no one knows what I am talking about or if it's just that no one can find a fault with my reasoning. Please please please could you post a reply...
  26. Rasalhague

    What Are Hamel & Schauder Bases and How Do They Relate to Vector Spaces?

    I'm trying to get a handle on these concepts, which are new to me. http://planetmath.org/encyclopedia/HamelBasis.html : Kreyszig offers the following example of a Schauder basis for l^2: e_i=(\delta_{ij}) (Introductory Functional Analysis with Applications, p. 43-44). Since any vector can...
  27. L

    Neutralization reaction: NaCl + H2O

    This is a subject I am unfamiliar with, and was just wondering where I should start in order to learn about them and how they affect different elements and form other substances step-by-step. From what I know, an Acid donated a Hydrogen Ion, and a Base takes a Hydrogen Ion, and an Acid has to...
  28. M

    Adding and subtracting different bases with common exponents in ring Z

    Homework Statement Calculate (5^7)-(7^7)+(9^7)-(11^7) in Mod24 Homework Equations The Attempt at a Solution I added all the bases and got -4 (which I changed to 4), then I took 4^7 and ended up with 16,384. I divided 16,384 by 24 as many times as I could, which gave me an end...
  29. L

    How Does the Buffering Action of Weak Acids Change During Titration?

    Homework Statement I'm a little confused, let's say i do a titration curve for a weak acid, at some point there's a buffering range near the pKa. Like i know that if we add a strong base like NaOH to acetic acid in a water solvent, then the OH- will combine with the H+ (hydronium ion) which...
  30. K

    Integration of bases other than e

    Find the integral ∫(3^(2x))/(1 + 3^(2x)) dx so I have that u= 1+3^(2x) and du=2ln3[3^(2x)] I=1/(2ln3)∫3^(2x)(2)(ln3)/(1 + 3^(2x)) dx Can anyone help me solve this please?
  31. H

    Can Excel be Used to Solve Logarithmic Equations with Different Bases?

    Homework Statement 9 log3(2x+1) + 4 log2(x+3) = 85. Find x! Homework Equations logarithm The Attempt at a Solution I have tried few things but useless. Do not know what to do with different bases. I also tried to change the RHS: 85 = 5 x 17 but nothing worked. Thank you very much
  32. S

    Linear algebra change of bases

    Homework Statement I'm working on the problems attached. Homework Equations The Attempt at a Solution I think I understand how to do problem 4.3 but I'm lost on 4.1 and 4.2, and I'm pretty sure i need to figure those out before I can do 4.3. Can someone please try to guide...
  33. A

    Finding the Bases for kernel and range of linear transformation.

    Homework Statement How do I find the bases for both the kernel and range of this linear transformation? Let T: R4 ----> R4 be the linear transformation that takes [1101] and [1011] to [2304] and takes [1110] and [0111] to [3120] a. Find the bases for both the kernel and the range of...
  34. B

    Infinite-dimensional vector spaces and their bases

    I was working on a problem earlier today and I didn't know the following result: Let S be a subset of an infinite-dimensional vector space V. Then S is a basis for V if and only if for each nonzero vector v in V, there exists unique vectors u1,u2,...,un in S and unique nonzero scalars...
  35. N

    Any evidence that some organisms use bases other than A T C & G?

    Yet another question about the evolution of life, likely in the Archaen or Proterozoic eons. As I understand it, the genomes of all life - archaea, bacteria, eukaryotes - is based on DNA, even that of eukaryote organelles, such as plastids and mitochondria. Are there any counter-examples, in...
  36. N

    Differential Geom.: Dual Space Bases for Beginners

    I'm a beginner at differential geometry. I have a problem about dual space. I understand why we use \left\{\frac{\partial}{\partial x^{\mu}}\right\} as the bases in vector space, but I have no idea why we use \left\{ dx^{\mu} \right\} as the bases of dual space. What is the reason of...
  37. F

    Image and nullspace bases of a linear transformation

    Homework Statement Let T be the linear transformation T: M2x2-->M2x2 given by T([a,b;c,d]) = [a,b;c,d][0,0;1,1] = [b,b;d,d] Find bases (consisting of 2x2 matrices) for the image of T and the nullspace of T. Homework Equations Standard basis of a 2x2 matrix...
  38. W

    Equivalent bases for Vector Spaces over Complexes.

    Hi: Given a fin.dim vector space V over R, and two different bases B_V,B_V' for V , we say that B_V,B'_V are equivalent as bases ( or have the same orientation) , if there exists a matrix T with TB=B', and DetT>0. How do we define equivalent bases for vector spaces over the...
  39. C

    Proof about bases for subspaces

    Homework Statement Prove that all bases for subspace V of R\hat{}N contain the same number of elements. Homework Equations The Attempt at a Solution I have absolutely no idea where to start this proof. Do I need to do something with finding an equation of the subspace, or not?
  40. S

    What is the point in calculating orthonormal bases for R3

    I'm just learning linear algebra, where we are taking a set of basis vectors and then using gram schmidt to convert them to an orthonormal basis. So I know I can do it. I know orthonormal is great. But as I'm modifying the basis anyway from the original, why don't I just forget the original...
  41. P

    Convert Numbers to Rational Bases: An Example

    I have been playing around with numbers in different bases and then I thought, what if they were in fractional bases. I found a way to convert numbers to fractional bases and have been searching on the internet and not found a similar way to do this. Anyway, here is an example of how I would do...
  42. H

    Can Commuting Operators in a Hilbert Space Share a Common Set of Eigenvectors?

    Given a set of n<d commuting operators, either degenerate or non-degenerate, in a d-dimensional Hilbert space, is there an effective analytical method of finding an orthonormal basis composed of d eigenvectors common to all the operators in the set? The operators are dxd complex square matrices...
  43. M

    Linear Algebra - Change of Bases

    Hi, again another problem: Let B1 = {( \stackrel{1}{3}),( \stackrel{1}{2})} and B_{2} = [ \frac{1}{\sqrt{2}}( \stackrel{1}{1}), \frac{1}{\sqrt{2}} (\stackrel{-1}{1}) ] Determine the representing matrix T = K_{B_{2},B_{1}} \in \Re^{2\times2} for the change from B1 coordinates to...
  44. A

    Conjugate Bases of NH2-: Identifying

    Homework Statement Indicate the conjugate bases of the following: NH2- NH2- Homework Equations The Attempt at a Solution This is the only information given. Can I assume that these species react with water? The ionic signs indicate that they are bases, but are they the...
  45. N

    Triangular numbers - Roots with bases other than 1

    Not sure where this should go, but - How would you calculate the square root of a triangular number with a base other than 1? For example, 2, 6, 12, 20 (Base 2). Would rather have help to figure it out than the actual formula.
  46. H

    Linearly Independent Sets and Bases

    Homework Statement So I'm trying to find a basis for the space that is spanned by the given vectors. {(1,0,0,1) (-2,1,-1,1) (6,-1,2,-1) (5,-3,3,-4) (0,3,-1,1)} These are written as column vectors. Homework Equations None really (that I know of) The Attempt at a Solution So I...
  47. T

    Basis of range of a matrix relative to some bases

    Homework Statement Let T be the linear transformation of R5 into R3 that has the matrix A = 1 3 2 0 -1 2 6 4 6 4 1 3 2 2 1 relative to the bases [(1,1,1,1,1), (1,1,1,1,0), (1,1,0,0,0), (1,0,0,0,0), (0,0,0,0,1)] of R5 and [(1,1,1), (0,1,0), (1,0,0)] of R3. Find a basis for the range...
  48. G

    Solving Transformation Matrix and Basis Problem

    I'm having some trouble understanding basis and how they relate to transformation matrices. Homework Statement Let e_1,e_2,e_3,e_4 be a basis in a four dimensional vector space V. Suppose that the linear transformation F on V has the matrix representation: [1 0 2 1;-1 2 1 3;1 2 5 5;2 -2 1...
  49. M

    Solving Log Equations (Different Bases)

    Homework Statement Only the equation: 2(3^x) = 7(5^x) Homework Equations N/A The Attempt at a Solution Okay. I've been fine learning the loglaws, until I hit this question. If they were the same base (or at least near the same, ex: 2 & 8) then I could simply change them into...
  50. I

    Calculating pH of .40M NH4ClO4 Stuck!

    Homework Statement I was asked to calculate the pH of .40M solution of NH4ClO4 but I wasn't given a Ka for it. Homework Equations The Attempt at a Solution I dissciated NH4ClO4 into: NH4(plus) and ClO4(minus) but I"m stuck there. What am I doing wrong?
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