linear algebra

  1. A

    I Meaning of each member being a unit vector

    Summary: Meaning of each member being a unit vector, and how the products of each tensor can be averaged. Hello! I am struggling with understanding the meaning of "each member is a unit vector": I can see that N would represent the number of samples, and the pointy bracket represents an...
  2. christang_1023

    Decompose Involutory Matrix into Difference of Two Idempotents

    I feel confused about proving the two terms are idempotents.
  3. T

    Simplifying a matrix algebra equation (revised)

    I have a matrix equation (left side) that needs to be formatted into another form (right side). I've simplified the left side as much as I could but can't seem to get it to the match the right side. I am unsure if my matrix algebra skills are lacking or if I somehow messed up the starting...
  4. S

    Basis of eigenvectors

    Okay so I found the eigenvalues to be ##\lambda = 0,-1,2## with corresponding eigenvectors ##v = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix} ##. Not sure what to do next. Thanks!!!
  5. F

    Python Invert a matrix from a 4D array : equivalence or difference with indexes

    I have a 4D array of dimension ##100\text{x}100\text{x}3\text{x}3##. I am working with `Python Numpy. This 4D array is used since I want to manipulate 2D array of dimensions ##100\text{x}100## for the following equation (it allows to compute the ##(i,j)## element ##F_{ij}## of Fisher matrix) ...
  6. E

    Expectation value of operators and squeezing in the even cat state

    I started and successfully showed that the expectation of X_1 and X_2 are zero. However the expectation value of X1^2 and X2^2 which I am getting is <X1^2> = 0.25 + \alpha^2 and <X2^2> = 0.25. How do I derive the given equations?
  7. bluesky314

    I Question about an eigenvalue problem: range space

    A theorem from Axler's Linear Algebra Done Right says that if 𝑇 is a linear operator on a complex finite dimensional vector space 𝑉, then there exists a basis 𝐵 for 𝑉 such that the matrix of 𝑇 with respect to the basis 𝐵 is upper triangular. In the proof, he defines U=range(T-𝜆I) (as we have...
  8. M

    Proof a property for a 3x3 matrix

    Let a 3 × 3 matrix A be such that for any vector of a column v ∈ R3 the vectors Av and v are orthogonal. Prove that At + A = 0, where At is the transposed matrix.
  9. RikaWolf

    I Linear Algebra - Inner Product problem

    I need help to know if I'm on the right track: Prove/Disprove the following: Let u ∈ V . If (u, v) = 0 for every v ∈ V such that v ≠ u, then u = 0. (V is a vector-space) I think I need to disprove by using v = 0, however I'm not sure.
  10. synMehdi

    I Linear least squares regression for model matrix identification

    Summary: I need to Identify my linear model matrix using least squares . The aim is to approach an overdetermined system Matrix [A] by knowing pairs of [x] and [y] input data in the complex space. I need to do a linear model identification using least squared method. My model to identify is a...
  11. U

    Vector space has dimension less than d

    1. Homework Statement Problem given to me for an assignment in a math course. Haven't learnt about roots of unity at all though. Finding this problem super tricky any help would be appreciated. Screenshot of problem below. 2. Homework Equations Unsure of relevant equations 3. The...
  12. GlassBones

    Isomorphisms preserve linear independence

    1. Homework Statement Let ##T:V \rightarrow W## be an ismorphism. Let ##\{v_1, ..., v_k\}## be a subset of V. Prove that ##\{v_1, ..., v_k\}## is a linearly independent set if and only if ##\{T(v_1), ... , T(v_2)\}## is a linearly independent set. 2. Homework Equations 3. The Attempt at a...
  13. A

    A The product of a matrix exponential and a vector

    Hello everybody! I was studying the Glashow-Weinberg-Salam theory and I have found this relation: $$e^{\frac{i\beta}{2}}\,e^{\frac{i\alpha_3}{2} \begin{pmatrix} 1 & 0 \\ 0 & -1 \\ \end{pmatrix}}\, \frac{1}{\sqrt{2}}\begin{pmatrix} 0\\ v \\ \end{pmatrix} =...
  14. TachyonLord

    B How do I improve my proof writing style?

    So I've taken this Linear Algebra class as an elective. So there's stuff that is so obvious and logically/analytically easy to prove but I honestly don't understand how to prove them using the standard way. So what should I do about this ? And I really like linear algebra so I don't want to mess...
  15. S

    I Can't understand a step in an LU decomposition proof

    I'm reading about the LU decomposition on this page and cannot understand one of the final steps in the proof to the following: ---------------- Let ##A## be a ##K\times K## matrix. Then, there exists a permutation matrix ##P## such that ##PA## has an LU decomposition: $$PA=LU$$ where ##L## is a...
  16. GlassBones

    How to show a subspace must be all of a vector space

    1. Homework Statement Show that the only subspaces of ##V = R^2## are the zero subspace, ##R^2## itself, and the lines through the origin. (Hint: Show that if W is a subspace of ##R^2## that contains two nonzero vectors lying along different lines through the origin, then W must be all of...
  17. GlassBones

    Formulation of a proof of subspaces

    1. Homework Statement Let W be a subspace of a vector space V, let y be in V and define the set y + W = \{x \in V | x = y +w, \text{for some } w \in W\} Show that y + W is a subspace of V iff y \in W. 2. Homework Equations 3. The Attempt at a Solution Let W be a subspace of a vector space...
  18. Essence of linear algebra series -chapter 1 - 3blue1brown

    Essence of linear algebra series -chapter 1 - 3blue1brown

    The first video In 3blue1brown’s essence of linear algebra series.
  19. F

    A How to measure the first qubit in two qubit system? QC

    I was wondering how to measure the first or even the second qubit in a quantum computing system after for example a Hadamard Gate is applied to the system of these qubits: A|00>+B|01>+C|10>+D|11>? A mathematical and intuitive explanation would be nice, I am a undergraduate sophomore student...
  20. G

    Reduced equation of quadratic forms

    1. Homework Statement Given the following quadric surfaces: 1. Classify the quadric surface. 2. Find its reduced equation. 3. Find the equation of the axes on which it takes its reduced form. 2. Homework Equations The quadric surfaces are: (1) ##3x^2 + 3y^2 + 3z^2 - 2xz +...
  21. Destroxia

    I DSP: Recurrence Relations in a Linear Algebra Equation

    Hello, I've been working through some Digital Signal Processing stuff by myself online, and I saw a system that I wanted to write down as a Linear Algebra Equation. It's a simple delay feedback loop, looks like this: The (+) is an adder that adds 2 signals together, so the signal from x[n]...
  22. SebastianRM

    I Description of Inner Product

    Hey, I am currently reading over the linear algebra section of the "introduction to quantum mechanics" by Griffiths, in the Inner product he notes: "The inner product of two vector can be written very neatly in terms of their components: <a|B>=a1* B1 + a2* B ... " He also took upon the...
  23. O

    Change of basis computation gone wrong...

    1. Homework Statement Consider the real-vector space of polynomials (i.e. real coefficients) ##f(x)## of at most degree ##3##, let's call that space ##X##. And consider the real-vector space of polynomials (i.e. real coefficients) of at most degree ##2##, call that ##Y##. And consider the...
  24. T

    Number of indie vectors ##\leq ## cardinality of spanning set

    1. Homework Statement In a finite-dimensional vector space, the length of every linearly independent list of vectors is less than or equal to the length of every spanning list. It's quite long :nb), hope you guys read through it. Thanks! :smile: 2. Homework Equations N/A 3. The Attempt at a...
  25. I

    Projections and direct sum

    1. Homework Statement Let ##V = \mathbb{R}^4##. Consider the following subspaces: ##V_1 = \{(x,y,z,t)\ : x = y = z\}, V_2=[(2,1,1,1)], V_3 =[(2,2,1,1)]## And let ##V = M_n(\mathbb{k})##. Consider the following subspaces: ##V_1 = \{(a_{ij}) \in V : a_{ij} = 0,\forall i < j\}## ##V_2 =...
  26. LarryC

    Simultaneous Diagonalization for Two Self-Adjoint Operators

    (a) and (b) are fairly traditional, but I have trouble understanding the phrasing of (c). What makes the infinite dimensionality in (c) different from (a) and (b)?
  27. CharlieCW

    Transforming one matrix base to another

    1. Homework Statement The SO(3) representation can be represented as ##3\times 3## matrices with the following form: $$J_1=\frac{1}{\sqrt{2}}\left(\matrix{0&1&0\\1&0&1\\ 0&1&0}\right) \ \ ; \ \ J_2=\frac{1}{\sqrt{2}}\left(\matrix{0&-i&0\\i&0&-i\\ 0&i&0}\right) \ \ ; \ \...
  28. D

    I HHL Algorithm for Solving Linear Equations

    I have a question about HHL algorithm for solving linear equations of the form: A x = b Where A, x and b are matrices Take for example 4x1 + 2x2 =14 5x1 + 3x2 = 19 HHL apply the momentum operator eiAτto/T on the state, do a Fourier Transform on |b> and...
  29. mcabbage

    Courses On the benefits of retaking advanced linear algebra

    I'm a physics student who has the option to take some advanced math courses (Real analysis through Rudin and beyond, functional analysis if I have time, as well as algebra through Artin). I'm only just going into my second year this term, and will either be retaking linear algebra 2, or taking...
  30. Prez Cannady

    I Confused by this result for the tensor product of two vectors

    Given two probability distributions ##p \in R^{m}_{+}## and ##q \in R^{n}_{+}## (the "+" subscript simply indicates non-negative elements), this paper (page 4) writes down the tensor product as $$p \otimes q := \begin{pmatrix} p(1)q(1) \\ p(1)q(2) \\ \vdots \\ p(1)q(n) \\ \vdots \\...