Linear algebra Definition and 999 Threads

Dear Physics Forum personnel, I am a college sophomore in U.S. with a major in mathematics and an aspiring algebraic number theorist. I wrote this email to seek a recommendation on one or two outstanding linear algebra textbook that can supplement the Linear Algebra (Friedberg et al.), which...

Hey guys, I'm hoping to get some advice on how to approach this course I'm currently enrolled in. I'm currently enrolled in linear algebra 1, while I think they're pretty generic for outlines, this is what my school says: Before the midterm I was doing the assignments and, for the most part...

So, I understand that the dual space, V^{\ast} of a vector space V over a scalar field \mathbb{F} is the set of all linear functionals f^{\ast}:V\rightarrow\mathbb{F} that map the vectors in V to the scalar field \mathbb{F}, but I'm confused as to the meaning of the term dual?! Is it just that...

Homework Statement T(a0 + a1t+a2t2) = 3a0 + (5a0 - 2a1)t + (4a1 + a2)t2 Find the image of the vectors : 1. 1 2. t 3. t2 Homework Equations T(a0 + a1t+a2t2) = 3a0 + (5a0 - 2a1)t + (4a1 + a2)t2 The Attempt at a Solution I don't know how my book solves these transformations, but the answers are...

Hi I was just curious if this method of solving whether or not two spheres intersect is a viable method that will give me the correct answer. Say if I am given the two equations of the sphere's is it viable to: Find the centre and radius of each sphere. Find the magnitude of the distance of...

Hey guys, I'm currently enrolled to take Linear Algebra in the fall, however, after talking with a friend in math, he recommended since it's quite a brutal course that I take it over the summer. Does anyone here think that's a terrible idea? My reasoning for asking is because here we don't...

Pre-knowledge If V and W are finite-dimensional vector spaces, and dim(V) does not equal dim(W) then there is no bijective linear transformation from V to W. An isomorphism between V and W is a bijective linear transformation from V to W. That is, it is both an onto transformation and a one...

Pre-knowledge A matrix is a linear transformation if, T(u+v)= T(u) +T(v) and T(cu)=cT(u). Theorem 8.4.2 If V is a finnite dimensional vector space, and T: V-> V is a linear operator then the following are equivalent. a) T is one to one, b) ker(T)=0, c)...

Homework Statement If matrix ## C = \left[ {\begin{array}{c} A \\ B \ \end{array} } \right]## then how is N(C), the nullspace of C, related to N(A) and N(B)? Homework Equations Ax = 0; x = N(A) The Attempt at a Solution First, I thought that the relation between A and B with C is ## C = A...

I'm looking for a book to self-study this summer before my last course of Bachillerato (A Levels or High School in other countries). My performance in Mathematics has only improved and I've just been given the maximum mark this last term (10/10 or A+). This has motivated me a lot to keep...

Suppose I have already found the surface normal vectors to a set of points (x,y), how do I compute the surface height z(x,y)? Basically what I have are the normal vectors at each point (x,y) on a square grid. Then I calculate the vectors u = (x+1,y,z(x+1,y)) - (x,y,z(x,y)) and v =...

Hi can anyone give me some hints with this question thanks A = \begin{pmatrix} 3 & -2 &1 & 0 \\ 1 & 6 & 2 & 1 \\ -3 & 0 & 7 & 1 \end{pmatrix} be a matrix for T:ℝ4→ℝ3 relative to the basis B = {v1, v2, v3, v4} and B'= {w1, w2, w3} v1 = \begin{pmatrix} 0 \\ 1 \\ 1 \\ 1 \end{pmatrix} v2 =...

I am doing a project on image processing and I need to solve the following set of equations: nx+nz*( z(x+1,y)-z(x,y) )=0 ny+nz*( z(x+1,y)-z(x,y) )=0 and equations of the boundary (bottom and right side of the image): nx+nz*( z(x,y)-z(x-1,y) )=0 ny+nz*( z(x,y)-z(x,y-1) )=0 nx,ny,nz is the...

hey all, I've begun my engineering degree and have been enjoying it thoroughly, my first semester naturally wasn't too hot, I've got around a 2.7 so far, but my school states calculus 1 as a prerequisite for linear algebra, I hardly went to class and flipped classes but consistently scored above...

\begin{cases} x+ 2y - z + w - t = 0 \\ x - y + z + 3w - 2t = 0 \end{cases} Add 1st to the 2nd: $$2x + y + w - t = 0 \\ y = -2x -w + t = 0$$ Substitute y in the 1st: ##x + 2x + w - t + 3w - 2t + z = 0 \\ z = 3x - 4w + 3t## Both z and y in terms of x,w,t. Writing using matrix form...

As a quick continuation of the question...what mathematics courses would you (preferably PhD or pursuing a PhD currently) recommend for an undergraduate? I'm interested in high energy/particle physics (I'm working in a lab this summer so we'll see how that goes) and cosmology (no actual...

Homework Statement X ={(x1,x2,x2 −x1,3x2):x1,x2 ∈R} f(x1,x2,x2 −x1,3x2)=(x1,x1,0,3x1) 1. Find a basis for X. 2. Find dim X. 3. Find ker f and I am f 4. Find bases for ker f and I am f 5. Is f a bijection? Why? 6. Find a diagonal matrix for f.Homework EquationsThe Attempt at a Solution 1. Put...

X ={(x1,x2,x2 −x1,3x2):x1,x2 ∈R} f(x1,x2,x2 −x1,3x2)=(x1,x1,0,3x1) 1. Find a basis for X. 2. Find dim X. 3. Find ker f and I am f 4. Find bases for ker f and I am f 5. Is f a bijection? Why? 6. Find a diagonal matrix for f. My attempt: 1. (1, 1, 0, 3) and (1, 2, 1, 6) 2. Dim X = 2 3. Ker f = 0...

Hello, I noticed that the solution of a homogeneous linear second order DE can be interpreted as the kernel of a linear transformation. It can also be easily shown that the general solution, Ygeneral, of a nonhomogenous DE is given by: Ygeneral = Yhomogeneous + Yparticular My question: Is it...

Let $F$ be the set of infinite sequences $(a_1,a_2,a_3...)$, where $a_i \in \Bbb{R}$ that satisfy $a_{i+3}=a_i+a_{i+1}+a_{i+2}$ This describes a finite-dimensional vector space. Determine a basis for $F$.

I am finishing up Calculus II, which is mostly just solving integrals using techniques and finding the convergence of a series. My school is offering Linear Algebra in the summer, but it will not be available in Fall. I personally feel math to be my strong suit, since I had an A for my Calc 1...

I'm a physics major taking Linear Algebra right now. It's been pretty boring but I'm doing well. I'm just wondering about which concepts from this subject should I really focus on understanding and knowing how to apply well? Obviously I'm focusing on everything and trying to receive and A in...

I am researching ways linear algebra is integrated into aerospace engineering (I know its alot). I am looking for specific ways. Any help would be appreciated.

Please have a look at the attached images.I am attempting a proof for the statement : The algebraic multiplicity of an eigen value λ is equal to dim null [T - λ I] dim V. Please advise me on how to move ahead. Apparently, I am at the final inference required for a proof but unable to move...

Apologies if this should be in homework section but I thought it best suited here. Been revising past papers but with no solutions. the following questions all require just a true or false answer. Any help or confirmation of my answers would be appreciated. 1 - every N x N matrix has N...

Homework Statement Find the domain, target space, image, rank and nullity of the linear transformation T(A)=Av, where v= (1, 2) and A is any 2×2matrix. Homework Equations The Attempt at a Solution I believe I know the domain (R2x2 vector space) and target space (R2), but I am not sure how to...

Homework Statement Homework Equations Given above The Attempt at a Solution I used polyfit, but my mean swuare errors are way bigger than they should be- don't see what is wrong with my code! My code is ugly btw, my apologies. %Hw 7 clear all close all y3=[1960; 1965; 1970; 1975; 1980...

Hi I got stuck at the following circuit problem which involves linear algebra, since I am not a physics major, I don't even have the basics to get started. Please shed some light, really appreciate it! Thanks a lot. 1. Homework Statement Consider a long chain of resistors wired up like this...

I am trying to learn the formalism of qm, so i am following the book linear algebra done right but is it worth it to study every proof? I mean what is the attitude to follow with such a proof oriented book to eventually have a solid basis in the libear algebra of qm?

Homework Statement Homework Equations A=LU, U^-1 * L^-1= A^-1 , U^-1 * L^-1 * U^-1 * L^-1 = A^-2, The Attempt at a Solution I used MATLAB and the relations: U^-1 * L^-1= A^-1 , U^-1 * L^-1 * U^-1 * L^-1 = A^-2, to find a solution I found U^-1*L^-1 , let =B...

I decide to self-study linear algebra. I have heard words about some good books on this subject such as Sheldon Axler's. Unfortunately his book is only loanable for 4 days in my university library. There is this book from S. Lang that I can borrow for one month, so what do you think about this...

Homework Statement Show that the members of the Lie algebra of SO(n) are anti-symmetric nxn matrices. To start, assume that the nxn orthogonal matrix R which is an element of SO(n) depends on a single parameter t. Then differentiate the expression: R.RT= I with respect to the parameter t...

Homework Statement Show that the given matrices are row equivalent and find a sequence of elementary row ops that will convert A into B. a = 2 0 -1 1 1 0 -1 1 1 b = 3 1 -1 3 5 1 2 2 0 Homework EquationsThe...

Homework Statement "Let L1 be the line having parametric equations : x = 2 - s, y = -1 + 2s, z = 1+s and L2 be the line: x = 1 +t, y = 2+ t, z =2t . a. Do the lines intersect? If so, find the point of intersection. b. Find the point P on the graph of L1 that is closest to the graph of L2...

The question word for word : "Write the equation satisfied by all of the points P(x, y, z) that are at the same distance from the point F(0, 0, 4) and the plane z = 0." I figured I could maybe start by finding the distance between point F and the plane z = 0 but I can't figure out how to...

Hi, Im actually doing some systems theory and it requires some basic linear algebra stuff that I totally forgotten. Anyway:according to my prof. : For any matrix F. Im(BF) is contained in Im(B). here is my question: so Im(FB) is still in Im(B)? or is it true only when we multiply a matrix to...

Could anybody link me to some good examples on how to go about doing them? I honestly have no idea how to go about doing these two types of problems.

Homework Statement How do you know if say [(x_1,y_1),(x_2,y_2)] = x_1x_2 + 7y_1y_2 ? or any other equation? Homework EquationsThe Attempt at a Solution

First, let me say that I am a senior physics undergrad. I have failed Linear Algebra once before. Otherwise I am a straight A student. I'm also taking Ordinary Differential Equations right now, and I breeze through that class without a care in the world. I'm not sure if I've developed some sort...

Homework Statement What is the relation between the image of A and the image of A2 + A? Homework EquationsThe Attempt at a Solution im (A^2 + A) for x (A^2+A) is within the image. Linear combination properties show A^2 x + A x. Not sure where to go from here

Homework Statement Consider a 2x2 matrix A with A2=A. If vector w is in the image of A, what is the relationship between w and Aw? Homework Equations Linear transformation T(x)=Ax Image of a matrix is the span of its column vectors The Attempt at a Solution I know that vector w is one of the...

Homework Statement Prove that if A, B, and C are square matrices and ABC = I, then B is invertible and B−1 = CA. Homework EquationsThe Attempt at a Solution I think I have this figured out, just checking it. Heres what I got: ABC=I (ABC)B-1=IB-1 (B*B-1)AC=IB-1 I*AC=IB-1 Cancel I using left...

Hi, I that <I|M|J>=M_{I}^{J} is just a way to define the elements of a matrix. But what is |I>M_{I}^{J}<J|=M ? I don't know how to calculate that because the normal multiplication for matrices don't seem to work. I'm reading a book where I think this is used to get a coordinate representation of...

Homework Statement Prove that if A is an n x n matrix with the property A3=A, then det(A)=-1, det(A)=0, or det(A)=1 Homework EquationsThe Attempt at a Solution At first I started with the property A3=A I then applied the determinant to both sides. From this point I don't really see any...

Recommend a self study book for linear algebra with complex numbers

How do I prove that this problem doesn't have a solution? 1. Homework Statement 12. Consider the Leontief system Here the column sums are 1, violating the Leontief input constraint given in the text. Show that this system cannot have a solution. Hint: Add the three equations together...

Homework Statement The vectors that are perpendicular to (1,1,1) and (1,2,3) lie on a ____. Homework Equations The Attempt at a Solution This is really straight forward, but I cannot validate the answer to myself. The textbook says that they should lie on a line, but why is this? Obviously if...

I have the feeling that it is, but I am not really sure how to start the proof. I know I have to prove both closure axioms; u,v ∈ W, u+v ∈ W and k∈ℝ and u∈W then ku ∈ W. Do I just pick a vector arbitrarily say a vector v = (x,y,z) and go from there?

Hi all, I am taking Quantum Mechanics II this coming spring semester. However, I haven't taken Linear algebra yet but I don't want to take the class just because for some few topics. What linear algebra topics should I learn before the class? Thanks!

My question is as it says in the title really. I've been reading Nakahara's book on geometry and topology in physics and I'm slightly stuck on a part concerning adjoint mappings between vector spaces. It is as follows: Let W=W(n,\mathbb{R}) be a vector space with a basis...