# What is Linear algagbra: Definition and 22 Discussions

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1. ### Operator algebra: Hermicity and Eigenstates

A. I can show that A is either hermitian or antihermitian by $$(B^\dagger B=1-A^2)^\dagger$$ $$B^\dagger B=1-A^\dagger A^\dagger$$ comparing, we know that $$A^\dagger = \pm A$$ I don't know how I can make use of the communtation relation to get hermiticity of B. But I know that A and B must have...
2. ### Parametric curve question (determining unknown point)

My work so far: I am stuck because when I inputted the two possible values of t and k, neither solution worked. Where did I go wrong? Pointers would be appreciated! :)
3. ### I Are linear automorphisms nothing but the identity mapping?

Hi Physics Forums, Quick question! Are every automorphism on a vectors space ## V ## over some field ## \mathbb{F} ## nothing but the identity mapping in disguise? The reason for asking is; automorphisms are (from my point of view) basically a change of basis, and vectors are invariant under...
4. ### I Question on proof ##\Lambda^{\perp}(AU) = U^{-1} \Lambda^{\perp}(A)##

Say we have as special lattice ## \Lambda^{\perp}(A) = \left\{z \in \mathbf{Z^m} : Az = 0 \in \mathbf{Z_q^n}\right\}##. We define ##U \in \mathbf{Z^{m \times m}}## as an invertible matrix then I want to proof the following fact: $$\Lambda^{\perp}(AU) = U^{-1} \Lambda^{\perp}(A)$$ My idea: Let...
5. ### Prerequisite mathematics for intermediate mechanics?

I will be taking intermediate mechanics next semester, and am a bit concerned about potential gaps in my mathematical knowledge. Long story short, I used to be a physics major, switched to electrical engineering, and then decided to double major after a semester in EE. The issue is that, as a...
6. ### [Linear Algebra] Matrix Transformations

Evening, The reason for this post is because as the title suggests, I have a question concerning matrix transformation. These are essentially test prep problems and I am quite stuck to be honest. Here are the [questions](https://prnt.sc/riq7m0) and here are the...
7. ### Differential Equation with an Initial condition

Homework Statement x(dy/dx) = 3y +x4cos(x), y(2pi)=0 Homework Equations N/A The Attempt at a Solution I've tried a couple different ways to make this separable, but you always carry over a 1/dx or 1/dy term and I can never fully separate this. I've also tried to do a Bernoulli differential...
8. ### Find All Possible Values of x When 3 Vectors are Linearly Dependent

Homework Statement >There are three vector $$\vec a ,\vec b, \vec c$$ in three-dimensional real vector space, and the inner product between them $$\vec a . \vec a=\vec b.\vec b=\vec a.\vec c=1, \vec a.\vec b=0, \vec c.\vec c=4$$ When setting $$x = \vec b.\vec c$$ , (dot here means dot...
9. ### 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...
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### B Why do equations with two distinct variables with 2 distinct

My question is about basic algebra. I am thinking about the "why" here and I'm looking for an intuitive answer.If you have the following equations: =>S+U=90 =>S+U=90 and =>40S+25U=2625 =>40S+25U=2625 you can then rewrite S=90−U and then substitute. Now you have a single equation with one...
11. ### I Problem understanding the SPAN

Hello everyone, I'm a CS student and I'm taking a course called Linear Algebra it's very easy, but there is one thing that I'm not clearly understanding i know how the general way to prove if given vectors span a vspace, ex : v1,v2,v3 i put them in a Matrix form and prove the determinant...
12. ### A Find vectors in Orthogonal basis set spanning R4

An orthogonal basis set spanning R4 has four vectors, v1, v2, v3 and v4. If v1 and v2 are [ −1 2 3 0 ] and [−1 1 −1 0 ] ﬁnd v3 and v4. Please explain this in a very simple way.
13. ### Rounding error making my graphics barely off?

I'm trying to draw a circle and a (possibly rotated) square on a grid. I have the circle part down and it's the square that is giving me trouble. I am originally given 2 points which represent the coordinates containing opposite ends of the square. For example, those 2 points would be (8,14) and...
14. ### B is nonsingular -- prove B(transpose)B is positive definite.

Homework Statement Suppose B is a real nonsingular matrix. Show that: (a) BtB is symmetric and (b) BtB is positive definite 2. Homework Equations N/A The Attempt at a Solution I have managed to prove (a) by showing that elements that are symmetric on the diagonal are equal. However I have no...
15. ### Representing a transformation with a matrix

Homework Statement Use matrix multiplication to ﬁnd the 2×2 matrix P which represents projection onto the line y =√3x. Can you suggest another way of ﬁnding this matrix? Which vectors x∈R2 satisfy the equation Px = x? For which x is Px = 0? Homework Equations Dot product of vectors The...
16. ### I Linear algebra ( symmetric matrix)

I am currently brushing on my linear algebra skills when i read this For any Matrix A 1)A*At is symmetric , where At is A transpose ( sorry I tried using the super script option given in the editor and i couldn't figure it out ) 2)(A + At)/2 is symmetric Now my question is , why should it be...
17. ### Linear algebra - linear equation for a plane

The problem I am trying to write the equation for the plane on the following form ## ax + by + cz + d = 0 ## $$\begin{cases} x = 1 + s - t \\ y = 2 - s \\ z = -1 + 2s \end{cases}$$ The attempt ## s, t ## are the parameters for the two directional vectors which "support" the plane. ...
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### Courses Should I take linear algebra over the summer?

Current schedule: Summer: Elective Fall: -not important, but it can't be changed in any way- Spring: Differential Eqs., Classical Mechanics & Mathematical Methods 1 (Physics), Linear Algebra, Chemistry Lab, Elective I'm thinking that since the spring schedule looks kinda heavy I can switch up...
19. ### [Linear Algebra] Conjugate Transpose of a Matrix and vectors in ℂ

Homework Statement Let A be an n x n matrix, and let v, w ∈ ℂn. Prove that Av ⋅ w = v ⋅ A†w Homework Equations † = conjugate transpose ⋅ = dot product * = conjugate T = transpose (AB)-1 = B-1A-1 (AB)-1 = BTAT (AB)* = A*B* A† = (AT)* Definitions of Unitary and Hermitian Matrices Complex Mod...
20. ### All possible planes, given two points

Homework Statement Find the equation of all planes containing the points P(2, -1, 1) and Q(1, 0, 0) Homework EquationsThe Attempt at a Solution I use PQ to get a vector, (-1, -1, 1). I some how need to use another vector so I can use the cross product to find the planes. So i let another...
21. ### B What are the values in a vector?

I'm trying to understand the concept of vectors. Vectors have magnitude and a direction. When I read vector with some values \textbf{x} = \left(\begin{array}{c}x_1\\x_2\\x_3\end{array}\right) = \left(\begin{array}{c}1\\2\\3\end{array}\right) I'm not sure what these values are. Are the values...
22. ### How to triangulate two 3d lines

Hi guys, I am working on an computer vision project. the project uses two cameras to triangulate an object in front of the cameras. Homework Statement Express the object's location in 3d coordinates relative to the cameras. Homework Equations From the software i can get two line equations...