Linear algebra Definition and 999 Threads

Homework Statement Prove in general that if m does not equal n, then AB and BA cannot both be identity matrices, where A is mxn and B is nxm. Homework Equations None (that I know of at least). The Attempt at a Solution At first I thought it would be a good idea to define each...

Homework Statement Suppose T: V -> W is linear. Prove that T(0) = 0 The Attempt at a Solution T(v) = Av T(0) = A(0) = 0 Is that right?

Homework Statement So the question is, Prove the following: Let A be an n x n matrix. If there exists a vector v in Rn that is not a linear combination of the columns of A, then at least one column of A is not a pivot column. Homework Equations The only relevant theorem I think is the...

hello, the theorem says : let V be a vector space over the field K , let { v1 , v2 , ... , vm } be a basis of V over K let {w1 , w2 , ... , wn} be elements of V and assume that n is bigger than m , then { w1 , w2 , ... , wn } are linearly dependent the proof is written here but I...

Homework Statement Given A\in Mnxn and A = A2, show that C(A) +N(A) = ℝn. note: C(A) means the column space of A. N(A) means the null space of A Homework Equations These equations were proved in earlier parts of the problem... C(A) = {\vec{x}\in ℝn such that \vec{x} =...

Homework Statement If x = (x1, x2) and y = (y1, y2)... Show that <x,y> = 3(x1)(y1) - (x1)(y2) - (x2)(y1) + 3(x2)(y2) Homework Equations I know that to define it as an inner product space, the following must be correct: <x,y> = <y,x> a<x,y> = <ax,y> <x,y+z> = <x,y> + <x,z>...

Homework Statement Let A \in M_n(F) and v \in F^n . Also...[g \in F[x] : g(A)(v)=0] = Ann_A (v) is an ideal in F[x], called the annihilator of v with respect to A. We know that g \in Ann_A(v) if and only if f divides g in F[x]. f is the monic polynomial of lowest degree in the...

Homework Statement No idea how to solve this using graham schmidt. I know how to do graham schmidt and how to solve this problem if I didn't have to use graham schmidt, but I have no idea where to start in order to get my vectors to add to V Found c to be 87 by using vector...

Homework Statement Show that the system ∑ hereunder admits one unique solution ∑ = \left[\begin{array}{cc} 1 & a_{1} & a_{1}{}^{2} & a_{1}{}^{3} & | & b_{1}\\ 1 & a_{2} & a_{2}{}^{2} & a_{2}{}^{3} & | & b_{2}\\ 1 & a_{3} & a_{3}{}^{2} & a_{3}{}^{3} & | & b_{3}\\ 1 & a_{4} & a_{4}{}^{2} &...

Homework Statement Homework Equations Not sure. The Attempt at a Solution Have no idea, as I don't have any/much previous experience with Linear Algebra. Can anyone help me with starting on this, hints/tips?

Homework Statement Let A \in M_n(F) and v \in F^b . Also...[g \in F[x] : g(A)(v)=0] = Ann_A (v) is an ideal in F[x], called the annihilator of v with respect to A. We know that g \in Ann_A(v) if and only if f|g in F[x]. Let V = Span(v, Av, A^2v, ... , A^{k-1}v).. V is teh smallest...

Homework Statement Find an equation of the plane that has y-intercept -5 and is parallel to the plane containing the points P(3, -1, 2), Q(0, 2, 1) and R(5, 2, 0)Homework Equations ax + by + cz + d = 0 The Attempt at a Solution I got two directional vectors u = PQ = (-3, 3, -1) v = PR = (2...

Hi Guys I just got some problem about coding theory and I don't quite understand what question 2 is asking. Can you guys help me? Thanks a lot.

Homework Statement Let A \in M_n(F) and v \in F^n. Let k be the smallest positive integer such that v, Av, A^2v, ..., A^kv are linearly dependent. a) Show that we can find a_0, ... , a_{k-1} \in F wiht a_0v + a_1Av + ... + a_{k-1}A^{k-1}v + A^kv = 0 (note that teh coefficient of...

Homework Statement True or False: The linear combinations a_{1}v_{1} + a_{2}v_{2} and b_{1}v_{1} + b_{2}v_{2} can only be equal if a_{1} = b_{1} and a_{2} = b_{2} Homework Equations The Attempt at a Solution I have determined that this statement is false if at least of the...

Homework Statement Let E = {“ax+by+cz = d” | a; b; c; d ∈ R} be the set of linear equations with real coefficients in the variables x, y and z. Equip E with the usual operations on equations that you learned in high school. addition of equations, denoted below by “⊕” and multiplication by...

Homework Statement A is a mxn. V is nxn and invertible. Show that imA=imAV2. The attempt at a solution Up until now I haven't done much in the way of proving things. In this case is it enough to show that they are each closed under addition and scalar multiplication? Would that mean that imA is...

I am using An Introduction to numerical linear algebra by Charles Cullen and I'm not very satisfied with it. Kindly suggest me some alternatives. Also suggest good linear algebra book to clear up basics. finally also suggest any online study materials, lecture notes, videos regarding the...

Homework Statement Suppose v_1,v_2,v_3,...v_n are vectors such that v_1 does not equal the zero vector and v_2 not in span{v_1}, v_3 not in span{v_1,v_2}, v_n not in span{v_1,v_2,...v_(n-1)} show that v_1,v_2,v_3,...,V_n are linearly independent. Homework Equations linear independence...

Linear Algebra Homework help! Homework Statement Suppose a particular object is modeled as moving in an elliptical orbit centered at the origin. Its nominal trajectory is described in rectangular coordinates (r;s) by the constraint equation x1r^2 +x2s^2 +x3rs = 1, where x1; x2; and x3 are...

Homework Statement Let A \in M_n(F) and v \in F^n. Let v, Av, A^2v, ... , A^{k-1}v be a basis, B, of V. Let T:V \rightarrow V be induced by multiplication by A:T(w) = Aw for w in V. Find [T]_B, the matrix of T with respect to B. Thanks in advance Homework Equations...

I want to prove that: Ker(T*)=[Im(T)]^\bot Everything is in finite dimensions. What I'm trying: Let v be some vector in ImT, so there is v' so that Tv'=v. Let u be some vector in KerT*, so T*u=0. So now: <u,v>=<u,Tv'>=<T*u,v'>=0 so every vector in ImT is perpendicular to every vector...

Homework Statement Bonus] Let E = {“ax+by+cz = d” | a; b; c; d ∈ R} be the set of linear equations with real coefficients in the variables x, y and z. Equip E with the usual operations on equations that you learned in high school: addition of equations, denoted here by “⊕” and...

Homework Statement Which of the following are subspaces of F[R] = {f |f:R-->R}? a) U = {f e F[R]|f(-1)f(1)=0 b) V = " |f(1)+f(2)=0 c) S = " |f(x)=f(-x) d) T = " |f(1)<= 0 Homework Equations The Attempt at a Solution I got S and V or c) and b), is that correct? I...

Homework Statement In the figure, let S be an inertial frame and let S' be another frame that is boosted with speed v along its x'-axis w.r.t. S, as shown. The frames are pictured at time t = t0 = 0: A) Find the Non-relativistic transformation (Galilean Transformation) between the two...

Homework Statement Consider the vector space F(R) = {f | f : R → R}, with the standard operations. Recall that the zero of F(R) is the function that has the value 0 for all x ∈ R: Let U = {f ∈ F(R) | f(1) = f(−1)} be the subspace of functions which have the same value at x = −1 and x = 1...

Homework Statement Find the Matrix M which represents the reflection about the line L given by the equation y=(1/2)x. By two methods: a) By writing the composition as a composition of rotations and reflections about the x-axis. Note that the line L makes an angle of pi/6 with the x-axis...

Homework Statement Consider the vector space F(R) = {f | f : R → R}, with the standard operations. Recall that the zero of F(R) is the function that has the value 0 for all x ∈ R: Let U = {f ∈ F(R) | f(1) = f(−1)} be the subspace of functions which have the same value at x = −1 and x = 1...

A is a square matrix n*n with the following properties: A*A=A and A not equal I (identity matrix). How to prove the following equation: (I+A)^-1=I+A/2 ?

Hi, Everyone It is difficult to find nice workable books for more advanced linear algebra. There are numerous publications and internet materials, few of them are workable to me. Interested topics: unitary and Hermitian matrices, Jordan (canonical) form, tridiagonal matrix, Sylvester...

Homework Statement problem didn't state, but I assume let V be a vector space: V = C^3 and scalar is C Homework Equations Define a non-zero linear functional T on C^3 such that T ((1, 1, 1)) = T ((1, 1, −1)) = 0 The Attempt at a Solution So let X1 = (1, 1, 1); X2 = (1, 1, -1); It...

What topics/chapters are covered in a typical linear algebra class? I am a physics/math major but I won't be able to take classes for a few years. I am trying to teach myself linear algebra so I can read physics textbooks. Thanks

Homework Statement Find A if (2A-1 - 3I)T = 2* \begin{pmatrix} -1 & 2\\ 5 & 4 \end{pmatrix} Homework Equations The Attempt at a Solution I have no idea if I'm even on the right track of solving this question... I simplified the right hand side down to \begin{pmatrix} -2 & 4\\ 10 & 8...

Homework Statement Determine the standard matrix for the linear operator defined by the formula below: T(x, y, z) = (x-y, y+2z, 2x+y+z) Homework Equations The Attempt at a Solution No idea

Homework Statement Let there be 3 vectors that span a space: { |a>, |b>, |c> } and let n be a complex number. If the operator A has the properties: A|a> = n|b> A|b> = 3|a> A|c> = (4i+7)|c> What is A in terms of a square matrix? Homework Equations det(A-Iλ)=0 The Attempt...

Homework Statement Prove for an operator A that det(e^A) = e^(Tr(A)) Homework Equations The Attempt at a Solution I have no idea how to start. Can someone give me a hint? In general the operator A represented by a square matrix, has a trace Tr(A) = Ʃ A (nn) where A (nn) is...

Homework Statement I have 32 bills in my wallet in the denominations $1, $5, and $10, worth $100 in total. How many of each denomination do I have? Homework Equations A= # $1 bills B= # $5 bills C= # $10 bills A+B+C = 32 1A+5B+10C = 100 The Attempt at a Solution So I...

Hello guys, can someone help me with this question please?

I have done some "proofs" before in calculus. At this moment I am required to write proofs for linear algebra and I find them highly unintuitive and confusing -- I often don't know where to begin or what to do. Can you guys leave some pointers, tips, advice, etc. for how to prove things...

This is just a general question. When a coefficient matrix for a linear system has a determinant equal to 0. That means the coefficient matrix does not have an inverse, thus the system does not have a unique solution. Is the above statement correct? What exact is a unique solution? Is...

Homework Statement L = lambda. Prove: d(A^-1)/dL = -(A^-1)(dA/dL)(A^-1) Homework Equations ? The Attempt at a Solution I did this as an analogy with function of numbers, but don't know how to extend this to matricies. for example: lets say A = f(L) d(f(L)^-1)/dL = -...

let V be the collection of the 2*3 matrices with a real entries such that V={[a11 a12 a13 : a21 a22 a23] | a11+a23 =1} determine whether the following vector space axioms holds (a) for all α ε V there exists (-α) such that α + (-α)=0(vector)

Hey all, I am trying to get a head start on Linear Algebra before i start taking classes in a couple weeks. I am about to go into my second year undergraduate and all i have behind my belt is calculus (single varialbe, multivariable, and vector analysis (curl, divergence, etc)). I am...

[b]1. i am given a matrix A= 1 0 2 1 1 1 3 1 2 3 8 -2 -3 3 -5 1 and then it asks why do I know that span {a1, a2, a3, a4} are a subset of R^4. Homework Equations The Attempt at a Solution Is it as easy as saying "because there 4 vectors?"

Homework Statement Problem 1: If A is an m x n matrix and Ax = 0 for all x ε ℝ^n, prove that A = O. If A and B are m x n matrices and Ax = Bx for all x ε ℝ^n, prove that A = B. (O is the 0 matrix, x is the vector x, and 0 is the 0 vector.)2. The attempt at a solution First off, I understand...

Homework Statement I have to find for which "a" an eigenvalue for the following system is 0. The system: 1 -1 1 -1 2 -2 0 a 1 Homework Equations My characterstic equation: (1-λ)(2-λ)(1-λ)+2a -(1-λ) -a = 0The Attempt at a Solution I then proceed: (1-λ)(λ2-3λ-2+a) = 0 but then I'm kind...

I was going through Linear Algebra which is recommended as a prerequisite to Quantum Mechanics. The topic of LA is vast and deep. So I wanted to know which (specific) topics of LA should be covered as a prerequisite to QM.

Hey everyone, I'm majoring in physics and will be starting my first year in the fall. I'm currently registered in honors linear algebra, but have been thinking that it might be beneficial to take the regular linear algebra course. I'm also in honors calculus, but I know I want to stay in that. I...

Homework Statement Is a set of n-tuples which must respect the conditions of closure under addition and closure under scalar multiplication a vector space or a vector subspace? That is, in a 3-dimensional space, are planes which pass by the origin considered to be subspaces of the 3-dimensinal...