Search results

  1. P

    Affine Functions

    Homework Statement I'm trying to show that every affine function f can be expressed as: f(x) = Ax + b where b is a constant vector, and A a linear transformation. Here an "affine" function is one defined as possessing the property: f(\alpha x + \beta y) = \alpha f(x) + \beta f(y)...
  2. P

    Jordan canonical form

    Homework Statement Are the operators specified by the matrices: A = \left[\begin{array}{ccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 2 \end{array}\right] B = \left[\begin{array}{ccc} 4 & 1 & -1 \\ -6 & -1 & -3 \\ 2 & 1 & 1 \end{array}\right] equivalent? Homework Equations See...
  3. P

    Continuous Functions, Vector Spaces

    Homework Statement Is the set of all continuous functions (defined on say, the interval (a,b) of the real line) a vector space? Homework Equations None. The Attempt at a Solution I'm inclined to say "yes", since if I have two continuous functions, say, f and g, then their sum f+g...
  4. P

    Norms of vectors problems

    This is the quick and, in some ways, "simple" way to get the answer. But I'm also thinking that most people don't see normed inner product spaces until they get to a course on real analysis, which might be a bit ahead of where the OP is at. But yes, you've basically laid out the right...
  5. P

    Norms of vectors problems

    Yes, correct. With inner products, we have: v \cdot w = |v| |w| \cos \theta where theta is the angle between the two vectors. In your original problem statement, you also indicated that: v \cdot w = \left(-\frac{1}{2}\right) |v| |w| Now, as a next step, using these two facts, are...
  6. P

    Norms of vectors problems

    Geometrically, the lengths of the vectors are involved in the dot product, but so is the angle between the two vectors. Do you know what this relationship is?
  7. P

    Norms of vectors problems

    Geometrically, what does the dot product represent?
  8. P

    Triple Integral in Spherical Coordinates

    The radius limits are correct. We start at 0, and move out to a radius of 2. For the angle limits, consider the following: Start at the origin (0,0,0) and move up the z-axis to 2: (0,0,2). Now, move the radius vector "down" (i.e, rotate it), until instead of pointing "up", it's now...
  9. P

    Triple Integral in Spherical Coordinates

    Your answer is close to what I get, but I would get different limits of integration. For the transformation, I would use: x = r \sin \theta \cos \phi y = r \sin \theta \sin \phi z = r\cos \theta If you work out the "integrand", I get: r^3 \cos \theta but then we're...
  10. P

    Triple Integral in Spherical Coordinates

    Do you know how to express x, y and z as functions of spherical coordinates? Start there....
  11. P

    Curl Test for vector fields

    Are you able to find the curl of this vector field? In general, if the curl of a vector field is zero, that means that (a) it's the gradient of a scalar (potential) function; and (b) the line integral of scalar potential function is path-independent.
  12. P

    Curl Test for vector fields

    What exactly do you mean by "curl test"?
  13. P

    Area in Polar Coordinates

    What you want to do is evaluate the double integral: A = \int \int dA where..... dA is as I've given above, right?
  14. P

    Area in Polar Coordinates

    Remember too, area must have units of "area".. that is to say "length*length".. So "dA" must have units of length*length. The unit of r is "length" The unit of dr is "length" The unit of dtheta is none, or dimensionless... You can't have dA = r^2drd\theta, b/c then you'd be left w/ units of...
  15. P

    Area in Polar Coordinates

    I'm thinking that in polar coordinates, you're looking at something more like: dA = rdr d\theta You should be able to integrate from there...
  16. P

    Area in Polar Coordinates

    Think about how "r" changes as "r" changes (I know that sounds silly)... And think about how "theta" changes as "theta" changes. So first think about r.. that's the simpler one. Suppose we go out to a distance "r" (in polar coordinates) from the center of origin. Then we go just a bit...
  17. P

    Existence and uniqueness

    I know you've already "solved" that part, but personally I don't like thinking of exactness in terms of "P's" and "Q's", unless those things have been carefully defined. What you're really trying to do w/ exactness suppose you have a solution to the D.E. given by: u(x,y) = c Then you...
  18. P

    Area in Polar Coordinates

    You'll end up w/ a double integral.. Something that looks like: A = \int \int dA(r,\theta) What is the differential element for the radius? What is the differential element for the angle? I don't want to give the whole answer away, but let's say that (this is somewhat sloppy...
  19. P

    Hyperplanes and Factor Spaces

    OK, so suppose we have a set H such that given x,y \in H, we know that \alpha x + (1-\alpha)y \in H for all alpha. The object is to prove that H is a hyperplane. Note that this equation can be rewritten as: y + \alpha(x-y) \in H So if we can prove that the difference between two...
  20. P

    Hyperplanes and Factor Spaces

    Homework Statement I'm working on a problem involving hyperplanes and factor spaces. It involves a bit of setup. I'll describe first the definitions. Suppose you have a vector space K, of dimension n. Suppose you have a linear subspace L of K. Choose a vector x_0 \in K, then the hyperplane H...
  21. P

    Linear Algebra Dimension Proof

    Yes, essentially. You mentioned in the OP something about a "rank-nullity" theorem. I'm not sure exactly what you meant about "rank-nullity" theorem, but yes, the rank of a matrix is the number of linearly independent columns (or rows), and so if the number of rows is k, then you must...
  22. P

    Linear Algebra Dimension Proof

    I wouldn't even mess around w/ an augmented matrix, or trying to actually, "physically" reduce the matrix into some kind of row-echelon form. Just consider the linear system: A \cdot x = 0 A is a k-by-n matrix.. that is, it has k rows and n columns. x is an n-dimensional vector. 0...
  23. P

    Linear Algebra Dimension Proof

    For the sake of illustration, suppose we are working in V = R^2, D=2, and suppose you choose S=3 vectors. You desire to prove that these three vectors must be linearly dependent in R^2. Form a linear combination of these vectors. You desire to verify whether or not a_1v_1 + a_2v_2 +...
  24. P

    Linear Independence of Polynomials

    Good point, but that may be ambiguity more to my use of the word "polynomial"... What I mean is simply powers of x raised to some real number r_i. After all, x^{-2} and x^{-3} are linearly independent also, yes? True, but we could just multiply by x^{r_1+1} to get back to the induction...
  25. P

    Linear Independence of Polynomials

    Homework Statement Given a set of polynomials in x: x^{r_1}, x^{r_2},...,x^{r_n} where r_i \neq r_j for all i \neq j (in other words, the powers are distinct), where the functions are defined on an interval (a,b) where 0 < a < x < b (specifically, x \neq 0), I'd like to show that this...
  26. P

    Differential equation

    Yes, so according to Mathematica the form works out to: \int \frac{dx}{x\sqrt{x-a}} = \frac{2}{\sqrt{a}}\tan^{-1}\left(\sqrt{\frac{x-a}{a}}\right) which is fine. I think that answers my question. I have another related question though. This whole question/thread is taken from a...
  27. P

    Differential equation

    My question is not so much about how to do the change of variables.. That I'm pretty sure I can do. My question is more that, once I do the change of variables, I get an expression like: \left(\frac{dr}{d\theta}\right)^2 = \frac{r^2(r-a)}{a} or, perhaps "simplifying"...
  28. P

    Differential equation

    In the original equation, differentiate y wrt x. That is: y' = \frac{dy}{dx}
  29. P

    Hyperbolic Equation, Elliptic Equation

    Homework Statement From the relation: A(x^2+y^2) -2Bxy + C =0 derive the differential equation: \frac{dx}{\sqrt{x^2-c^2}} + \frac{dy}{\sqrt{y^2-c^2}} = 0 where c^2 = AC(B^2-A^2) The Attempt at a Solution I'm able to (more or less) do the derivation, but I think the correct...
  30. P

    Differential equation

    Homework Statement Make the following change of variables: x = r \cos \theta y = r \sin \theta and integrate the following equation: (xy'-y)^2 = a(1+y'^2)\sqrt{x^2+y^2} The Attempt at a Solution First it's worth noting that the equation x^2+y^2=a^2 (even without changing...
Top