What is Linear algagbra: Definition and 21 Discussions

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! :)

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...

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...

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...

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...

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...

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...

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...

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...

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...

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.

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...

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...

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...

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...

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. $$...

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...

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...

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...

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...

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...