Linear algebra Definition and 999 Threads

  1. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 37: Matrix of a linear map - 1

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 37: Matrix of a linear map - 1

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  2. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 38: Matrix of a linear map - 2

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 38: Matrix of a linear map - 2

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  3. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 39: Matrix of a linear map - 3

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 39: Matrix of a linear map - 3

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  4. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 40: Change of bases

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 40: Change of bases

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  5. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 41: Computational rules for matrices

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 41: Computational rules for matrices

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  6. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 42: Rank of a matrix

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 42: Rank of a matrix

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  7. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 43: Computation of the rank of a matrix

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 43: Computation of the rank of a matrix

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  8. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 44: Elementary matrices

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 44: Elementary matrices

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  9. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 45: Elementary operations on matrices

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 45: Elementary operations on matrices

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  10. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 46: LR decomposition

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 46: LR decomposition

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  11. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 47: Elementary Divisor Theorem

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 47: Elementary Divisor Theorem

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  12. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 48: Permutation groups

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 48: Permutation groups

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  13. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 49: Canonical cycle decomposition of permutations

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 49: Canonical cycle decomposition of permutations

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  14. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 50: Signature of a permutation

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 50: Signature of a permutation

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  15. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 51: Introduction to multilinear maps

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 51: Introduction to multilinear maps

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  16. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 52: Multilinear maps continued

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 52: Multilinear maps continued

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  17. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 53: Introduction to determinants

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 53: Introduction to determinants

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  18. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 54: Determinants continued

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 54: Determinants continued

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  19. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 55: Computational rules for determinants

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 55: Computational rules for determinants

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  20. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 56: Properties of determinants and adjoint of a matrix

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 56: Properties of determinants and adjoint of a matrix

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  21. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 57: Adjoint determinant theorem

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 57: Adjoint determinant theorem

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  22. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 58: The determinant of a linear operator

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 58: The determinant of a linear operator

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  23. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 59: Determinants and Volumes

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 59: Determinants and Volumes

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  24. Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 60: Determinants and Volumes continued

    Linear Algebra by Prof. Dilip Patil (NPTEL):- Lecture 60: Determinants and Volumes continued

    COPYRIGHT strictly reserved to Prof. Dilip P. Patil and NPTEL, Govt. of India. Duplication prohibited. Lectures: http://www.nptel.ac.in/courses/111108098/ Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111108098
  25. I

    [Linear Algebra] Linear Transformations, Kernels and Ranges

    Homework Statement Prove whether or not the following linear transformations are, in fact, linear. Find their kernel and range. a) ## T : ℝ → ℝ^2, T(x) = (x,x)## b) ##T : ℝ^3 → ℝ^2, T(x,y,z) = (y-x,z+y)## c) ##T : ℝ^3 → ℝ^3, T(x,y,z) = (x^2, x, z-x) ## d) ## T: C[a,b] → ℝ, T(f) = f(a)## e) ##...
  26. I

    [Linear Algebra] Sum & Direct Sum of Subspaces

    ⇒Homework Statement [/B] Calculate ##S + T## and determine if the sum is direct for the following subspaces of ##\mathbf R^3## a) ## S = \{(x,y,z) \in \mathbf R^3 : x =z\}## ## T = \{(x,y,z) \in R^3 : z = 0\}## b) ## S = \{(x,y,z) \in \mathbf R^3 : x = y\}## ## T = \{(x,y,z) \in \mathbf R^3 ...
  27. I

    [Linear Algebra] Another question on subspaces

    Homework Statement Let ##V## be the vector space of the sequences which take real values. Prove whether or not the following subsets ##W \in V## are subspaces of ##(V, +, \cdot)## a) ## W = \{(a_n) \in V : \sum_{n=1}^\infty |a_n| < \infty\} ## b) ## W = \{(a_n) \in V : \lim_{n\to \infty} a_n...
  28. I

    [Linear Alg] Determining which sets are subspaces of R[x]

    Homework Statement [/B] Which of the following sets are subspaces of ##R[x]?## ##W_1 = {f \in \mathbf R[x] : f(0) = 0}## ##W_2 = {f \in \mathbf R[x] : 2f(0) = f(1)}## ##W_3 = {f \in \mathbf R[x] : f(t) = f(1-t) \forall t \in \mathbf R}## ##W_4 = {f \in \mathbf R[x] : f = \sum_{i=0}^n...
  29. I

    Determining if a subset W is a subspace of vector space V

    Homework Statement Let V = RR be the vector space of the pointwise functions from R to R. Determine whether or not the following subsets W contained in V are subspaces of V. Homework Equations W = {f ∈ V : f(1) = 1} W = {f ∈ V: f(1) = 0} W = {f ∈ V : ∃f ''(0)} W = {f ∈ V: ∃f ''(x) ∀x ∈ R} The...
  30. M

    Calculus Which books for Calculus AND Linear Algebra

    I wanted to go through Calculus and then Linear Algebra following either of two paths: a) Keisler's Infinitesmal approach>>>Nitecki Deconstructing Calculus>>>Nitecki Calculus in 3D>>>Freidberg's Linear Algebra OR b) Simmons Calculus with analytic geometry>>>Apostol Vol 1>>>>Apostol Vol...
  31. PcumP_Ravenclaw

    Linear Algebra Prerequisite of the book "Linear Algebra done right"

    Dear Fellows, I have recently completed the study of Stewart's calculus. Next, I want to read Linear Algebra. I have bought Sheldon Axler's "Linear Algebra done right" textbook. I want to know if my knowledge of calculus is enough to tackle this book or should I first...
  32. O

    Find the distance from the point P to a line - linear algebra

    Homework Statement Find the distance from point P (1,7,3) to the line (x,y,z) = (-2,1,4) + s(1,-3,4), s is a free variable Homework Equations projnQP = ( QP⋅n/(lengthQP)(lengthn) )(n) The Attempt at a Solution I'm not quite sure about how to find the normal (n) here, but if I make s=0, I'm...
  33. bornofflame

    [Linear Algebra] Show that H ∩ K is a subspace of V

    Homework Statement From Linear Algebra and Its Applications, 5th Edition, David Lay Chapter 4, Section 1, Question 32 Let H and K be subspaces of a vector space V. The intersection of H and K is the set of v in V that belong to both H and K. Show that H ∩ K is a subspace of V. (See figure.)...
  34. binbagsss

    Linear Algebra: 2 eigenfunctions, one with eigenvalue zero

    Homework Statement If I have two eigenfunctions of some operator, that are linearly indepdendent e.g ##F(x) , G(x)+16F(x) ## and ##F(x)## has eigenvalue ##0##, does this mean that ## G(x) ## must itself be an eigenfunction? I thought for sure yes, but the way I particular question I just...
  35. B

    Linear Algebra System of Equations/Rates Application Help

    Homework Statement Suppose that we have a system consisting of two interconnected tanks, each containing a brine solution. Tank A contains x(t) pounds of salt in 200 gallons of brine, and tank B contains y(t) pounds of salt in 300 gallons of brine. The mixture in each tank is kept uniform by...
  36. M

    Linear algebra, field morphisms and linear independence

    Homework Statement Let f1,f2, ..., fn : K -> L be field morphisms. We know that fi != fj when i != j, for any i and j = {1,...,n}. Prove that f1,f2, ..., fn are linear independent / K. Homework Equations f1, ..., fn are field morphisms => Ker (fi) = 0 (injective) The Attempt at a Solution I...
  37. Carson

    Determine if function forms a vector space

    Homework Statement Problem- Determine if the set of all function y(t) which have period 2pi forms a vector space under operations of function addition and multiplication of a function by a constant. What I know- So I know this involves sin, cos, sec, and csc. Also I know that a vector space...
  38. A

    I State Vectors vs. Wavefunctions

    Hi physicsforums, I am an undergrad currently taking an upper-division course in Quantum Mechanics and we have begun studying L^2 space, state vectors, bra-ket notation, and operators, etc. I have a few questions about the relationship between L^2, the space of square-integrable complex-valued...
  39. D

    I Can You Add a Scalar to a Matrix Directly?

    So, I recently came across this example: let us "define" a function as ƒ(x)=-x3-2x -3. If given a matrix A, compute ƒ(A). The soution proceedes in finding -A3-2A-3I where I is the multiplicative identity matrix. Now , I understand that you can't add a scalar and a matrix, so the way I see it is...
  40. bornofflame

    [Linear Algebra] Construct an n x 3 matrix D such that AD=I3

    Homework Statement Suppose that A is a 3 x n matrix whose columns span R3. Explain how to construct an n x 3 matrix D such that AD = I3. "Theorem 4" For a matrix A of size m x n, the following statements are equivalent, that is either all true or all false: a. For each b in Rm, Ax = b has a...
  41. S

    I Linear mapping of a binary vector based on its decimal value

    Given an ##N## dimensional binary vector ##\mathbf{v}## whose conversion to decimal is equal to ##j##, is there a way to linearly map the vector ##\mathbf{v}## to an ##{2^N}## dimensional binary vector ##\mathbf{e}## whose ##(j+1)##-th element is equal to ##1## (assuming the index starts...
  42. Euler2718

    I Clarifying a corollary about Quadratic Forms

    The question comes out of a corollary of this theorem: Let B be a symmetric bilinear form on a vector space, V, over a field \mathbb{F}= \mathbb{R} or \mathbb{F}= \mathbb{C}. Then there exists a basis v_{1},\dots, v_{n} such that B(v_{i},v_{j}) = 0 for i\neq j and such that for all...
  43. F

    Linear algebra matrix to compute series

    Post moved by moderator, so missing the homework template. series ##{a_n}## is define by ##a_1=1 ## , ##a_2=5 ## , ##a_3=1 ##, ##a_{n+3}=a_{n+2}+4a_{n+1}-4a_n ## ( ##n \geq 1 ##). $$\begin{pmatrix}a_{n+3} \\ a_{n+2} \\ a_{n+1} \\ \end{pmatrix}=B\begin{pmatrix}a_{n+2} \\ a_{n+1} \\ a_{n} \\...
  44. L

    Find the Fourier Series of the function

    Homework Statement Find the Fourier series of the function ##f## given by ##f(x) = 1##, ##|x| \geq \frac{\pi}{2}## and ##f(x) = 0##, ##|x| \leq \frac{\pi}{2}## over the interval ##[-\pi, \pi]##. Homework Equations From my lecture notes, the Fourier series is ##f(t) = \frac{a_0}{2}*1 +...
  45. L

    Identify the quadratic form of the given equation

    <Moderator's note: Moved from a technical forum and thus no template.> Hello I am given the following problem to solve. Identify the quadratic form given by ##-5x^2 + y^2 - z^2 + 4xy + 6xz = 5##. Finally, plot it. I cannot seem to understand what I have to do. The textbook chapter on...
  46. T

    Proving a statement about the rank of transformations

    Homework Statement How to prove ##max\{0, \rho(\sigma)+\rho(\tau)-m\}\leq \rho(\tau\sigma)\leq min\{\rho(\tau), \rho(\sigma)\}##? Homework Equations Let ##\sigma:U\rightarrow V## and ##\tau:V\rightarrow W## such that ##dimU=n##, ##dimV=m##. Define ##v(\tau)## to be the nullity of ##\tau##...
  47. L

    Inner Product, Triangle and Cauchy Schwarz Inequalities

    Homework Statement Homework Equations I am not sure. I have not seen the triangle inequality for inner products, nor the Cauchy-Schwarz Inequality for the inner product. The only thing that my lecture notes and textbook show is the axioms for general inner products, the definition of norm...
  48. L

    Linear Algebra - Find an orthogonal matrix P

    A problem that I have to solve for my Linear Algebra course is the following We are supposed to use Mathematica. What I have done is that I first checked that A is symmetric, i.e. that ##A = A^T##. Which is obvious. Next I computed the eigenvalues for A. The characteristic polynomial is...
  49. T

    I Geometric intuition of a rank formula

    I am trying to understand the geometric intuition of the above equation. ##\rho(\tau)## represents the rank of the linear transformation ##\tau## and likewise for ##\rho(\tau\sigma)##. ##Im(\sigma)## means the image of the linear transformation ##\sigma## and lastly, ##K(\tau)## is the kernel of...
  50. 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...
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