Linear algebra Definition and 999 Threads

Homework Statement Find the standard matrix representation for each of the following linear operators: L is the linear operator that reflects each vector x in R2 about the x1 axis and then rotates it 90° in the counterclockwise direction. Homework Equations The Attempt at a Solution So my...

Homework Statement Determine the kernel/range of each of the following linear operators on R3 L(X)=(x1,x1,x1)THomework Equations The Attempt at a Solution So first thing I did was create a 3x1 matrix filled with ones. I equaled it to zero and found x1=0 to be a solution. However I'm not...

Are there any resources consisting of a collection of problems on linear algebra for students to practice? I am looking for good interesting problems which test students’ understanding. These questions or examples should be for teaching rather than just testing. The level of difficulty is first...

Linear Algebra Problems Are there any resources consisting of a collection of problems on linear algebra for students to practice? I am looking for good interesting problems which test students’ understanding. These questions or examples should be for teaching rather than just testing. The...

Hello MHB, This is probably my first challenge problem which falls in the 'University Math' category. $V$ is a vector space over an infinite field $F$, prove that $V$ cannot be written as a set theoretic union of a finite number of proper subspaces.

Author: John Hubbard, Barbara Hubbard Title: Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach Amazon Link: https://www.amazon.com/dp/0971576653/?tag=pfamazon01-20

Homework Statement Assuming that all matrices are n\times n and invertible, solve for D. C^{T}B^{-1}A^{2}BAC^{-1}DA^{-2}B^{T}C^{-2}=C^{T} The Attempt at a Solution I tried to group all like terms and simplify. I'm pretty sure this is not allowed but I'm not really sure how to...

Homework Statement True/False: If true give a proof, if false give a counterexample. a) If A and B have the same eigenvector X, then A+B should also have the same eigenvector, X. b) if A has an eigenvalue of 2, and B has an eigenvalue of 5, then 7 is an eigenvalue of A+B...

Homework Statement Let A ε M2x2 prove χA(x) = x^2 -tr(A)x + det(A) Homework Equations The Attempt at a Solution Hi all, this is an assignment equation and the right hand side i can perfectly understand but i can't understand the left hand side, What is it i am looking for...

Homework Statement Suppose S,T ∈ L(V) and S is invertible. Prove that if p ∈ P(F) is a polynomial, then p(S*T*S-1)=S*p(T)*S-1.Homework Equations none The Attempt at a Solution Suppose by contradiction that for any p ∈ P(F), p(S*T*S-1)≠S*p(T)*S-1 for any p ∈ P(F). Since this is true for any...

I just had my first Linear Algebra at the Universidade Federal de Minas Gerais (Brazil). The teacher talked about vector spaces, and I found it very abstract. I am starting to think about the subject, and any comments are welcome! I just need a place to vent my thoughts and frustrations, so I am...

V is the direct sum of W1 and W2 in symbols: V = W1 (+) W2 if: V = W1 + W2 and W1 \cap W2 = {0} Show that the following statements are equivalent: 1. V = W1 (+) W2 2. Every vector v\inV can be written uniquely as w1 + w2 where w1\inW1 and w2\inW2 3. V = W1 + W2 and for...

Hello everyone, I have a linear algebra question regarding Cramer's rule. Homework Statement Using Cramer's rule, solve for x' and y' in terms of x and y. \begin{cases} x = x' cos \theta - y' sin \theta\\ y = x' sin \theta + y'cos \theta \end{cases} 2. Homework Equations ##sin^2 \theta...

Homework Statement For each of the following, give an example if it exists. If it doesn't exist, explain why. a) An invertible 3x3 matrix which is not diagonalizable. c) An 3x3 matrix A with A^6+I3=0 (Hint: Use the determinant) Homework Equations For a): I know in order for a...

1.Well this isn't a specific homework problem, I just need a quick idea to use for a paper. Our linear algebra professor is giving us a 50 point paper project/report that has to do with the applications of linear algebra in the field of study we have chosen to pursue. I would like to do...

Homework Statement S_1 and S_2 are subsets of a vector space. When is this: span(S_1 \cap S_2) = span(S_1) \cap span(S_2) true? Prove it.Homework Equations The Attempt at a Solution conjecture: iff the two subsets are vector spaces.

Homework Statement This should be an easy one, I'm just making sure that I'm not screwing up horribly. Prove that if v is in span(v1,v2, ..., vN), then v, v1, v2, ..., vN are linearly dependent.Homework Equations span(v1,v2, ..., vN) = {Ʃaivi}.The Attempt at a Solution If v is in span(v1,v2...

Homework Statement "Suppose A is an n x n matrix with the property that the equation Ax = b has at least one solution for each b in |Rn. Explain why each equation Ax = b has in fact exactly one solution." 3. Attempt at solution A*x=b => [Ax1 Ax2 Ax3 ... Axm] = [b1 b2 b3 ...bm], where the x...

Homework Statement I don't understand how line five counting from the top in the attached image. How does det (A^-1 det A) become (detA)^n? I get that the A^-1 was factored out but I don't get how (detA )= (detA)^n. Thank you... http://img28.imageshack.us/img28/8742/20130302114549.jpg...

Homework Statement Given v1, v2 ... vk and v, let U = span {v1, v2 ... vk} and W = span {v1, v2 ... vk, v}. Show that either dim W = dim U or dim W = 1 + dim U. The Attempt at a Solution I'm not really sure where to start. If I knew that {v1, v2 ... vk} was linearly independent, then it would...

Homework Statement Let {p, q} be linearly independent polynomials. Show that {p, q, pq} is linearly independent if and only if deg p ≥ 1 and deg q ≥ 1. Homework Equations λ1p + λ2q = 0 ⇔ λ1 = λ2 = 0 The Attempt at a Solution λ1p + λ2q + λ3pq = 0 I know if λ3 = 0, then the coefficients of...

Homework Statement Describe the possible echelon forms of the standard matrix for the linear transformation T. T: |R3 --> |R4 is one to one. The Attempt at a Solution T(x)=Ax. Right? So A must be the standard matrix. I got this: A = | £ * * | | 0 £ * | | 0 0 £ | | ? ? ? | Where £...

Can someone please recommend a linear algebra book that will be more visual friendly. I like to understand the concepts visually and the book that my class is using doesn't fit me too well. I am using Elementary Linear Algebra by Spence and Friedberg.

Homework Statement Describe how to create a coding matrix A so that A-1 has no fractions The answer is Multiply a few type III matrices together, so that det A = 1 What I put is to just use an upper triangular matrix, since the det of an upper triangular matrix is 1 and you have to do...

Hello, I am just not satisfied with the following theorem (I don't know it's name): Let T:R^n -> R^m be a linear transformation. Then T is one-to-one if and only if the equation T(x) = 0 has only the trivial solution. The "proof" involves saying that if T is not one-to-one, then there...

Homework Statement Let A be a square matrix. a. Show that (I-A)^-1 = I + A + A^2 + A^3 if A^4 = 0. b. Show that (I-A)^-1 = I + A + A^2 + ... + A^n if A^(n+1) = 0. Homework Equations n/a The Attempt at a Solution I thought I'd want to use the fact that the multiplication of a...

Homework Statement Let A and B be nxn matrices. Prove that if AB=I, then BA=I det(AB)=det(I) 1.det(A)*det(B)=1 det(BA)=det(I) 2.det(B)*det(A)=1 Equating 1 and two together I get det(B)*det(A)=det(A)*det(B) Thus AB=I, then BA=I Is this correct? Homework Equations...

Homework Statement Show that if the det(A)=1 then adj(adj(A))=A Given Goal det(A)=1 adj(adj(A))=A Using the following formula A-1=adj(A)/det(A) if det(A)=1 then A-1=adj(A) likewise A=adj(A-1)/det(A-1) if...

Homework Statement Let A and B be n × n matrices. Which of the following statements are always true? (i) If det(A) = det(B) then det(A − B) = 0. (ii) If A and B are symmetric, then the matrix AB is also symmetric. (iii) If A and B are skew-symmetric, then the matrix AT + B is also...

Homework Statement Given A=[1 2 1; 0 4 3; 1 2 2] determine the (2,3) entry of A-1 by computing a quotient of two determinants. This problem confused me a bit, do they just want us to divide the adj(A) by the det(A) in order which would give us A-1 and just state the (2,3) entry from...

Homework Statement Determine whether the following vectors are linearly independent in P3 "not sure what P3 stands for maybe polynomial of third degree?" 1,x2,x2-2 let p1(x)=1 p2(X)=x2 p3=x2-2 c1p1(x)+c2p2(x)+c3p3(x)=z where z=0x2+0x+0 I then create a matrix using the above...

Homework Statement Prove that if S is a subspace of R1, then either S={0} or S=R1. Trying to come up with a proof I dissected each statement, I know that in order for S to be a subspace the zero vector must lie within the subset. So I know S={0} is true. I then checked an arbitary...

Homework Statement Let x,y,and z be vectors in a vector space V. Prove that if x+y=x+z then y=z I know it can't be as simple as just subtracting vector x from both sides. What I'm thinking is out goal is to get y=z which means that there much be a zero vector in the relationship...

Homework Statement We are to show that the set C of complex numbers, with scalar multiplication dened by α(a + bi) = αa + αbi and addition defined by (a + bi) + (c + di) = (a + c) + (b + d)i, satises the eight axioms of a vector space I have a few questions about this problem...

Homework Statement Determine whether the following sets form subspaces of R^{2} A){(x_{1},x_{2})^{T} | x_{1}x_{2}=0} B){(x_{1},x_{2})^{T} | x_{1}=3x_{2}} Homework Equations checks: Does zero vector exist? Is the space closed under addition? Is the space closed under scalar multiplication?The...

http://i.imgur.com/MIazUji.png "Describe the matrix A so that Ax = [x1-x2; x2-x3;...;x(n-1) - x(n); x(n) - x1] Ax = b I feel like after getting the ball rolling I'll actually be able to work on this problem, but for the time being I haven't the slightest idea how to begin it...

Homework Statement Let T be a linear transformation from R3 to R3. Suppose T transforms (1,1,0) ,(1,0,1) and (0,1,1) to (1,1,1) (0,1,3) and (3,4,0) respectively. Find the standard matrix of T and determine whether T is one to one and if T is onto

This may sound like a dumb question, I heard that to understand Quantum Maths, I have to know Linear Algebra, Calculus, Differential Equations... I don't have any problems with Calculus and Differentials but Linear Algebra was a bit foggy sometimes... What are the topics in Linear Algebra that i...

Homework Statement I want to proove the determinant of the following 3x3 matrix is 0. 1 1 1 tanA tanB tanC tan2A tan2B tan2C where A+B+C=2pi...

Author: Otto Bretscher Title: Linear Algebra with Applications Amazon Link: https://www.amazon.com/dp/0136009263/?tag=pfamazon01-20 Table of Contents: Text Features Preface Linear Equations Introduction to Linear Systems Matrices, Vectors, and Gauss-Jordan Elimination On the...

What is the linear algebra background for Quantum maths? Matrices? Basis's? Matrix trans.? Coordinate Systems? Please help me, and I am sorry if i posted this in the wrong place...

Homework Statement Is U = {A| A \in nℝn, A is invertible} a subspace of nℝn, the space of all nxn matrices? The Attempt at a Solution This is easy to prove if you assume the regular operations of vector addition and scalar multiplication. Then the Identity matrix is in the set but 0*I and...

Homework Statement The question and my work is in the image it is pretty much to solve for X. Solve the following matrix equation Not quite sure how I keep messing this problem up. Homework Equations The Attempt at a Solution

Homework Statement Suppose we know that a linear system Ax = b has a unique solution. What can we say about the solutions of the linear system Ax = 0? a) It has the same solution. b) The solution to Ax = b is also a solution to Ax=0, but there may be other solutions. c) Ax = 0 has a...

hi, I understand how to do the rotation equation A = [ cosθ -sinθ sinθ cosθ] A*v = [ cosθ -sinθ * [ x = [ xcosθ - ysinθ sinθ cosθ] y ] xsinθ + ycosθ ] A*v = [ cos90 -sin90 * [ 6 sin90 cos90 ] 4 ] =...

Homework Statement http://i48.tinypic.com/2qu14ax.jpg Can you guys explain whether or not my proof would be sufficient. My thought process was to start with the definition. So if A is nonsingular is means that A has an inverse such that A=A-. I used that same thinking for B. B=B- Then...

Homework Statement A function f:[a,b] \rightarrow ℝ is called piecewise continuous if there exists a finite number of points a = x0 < x1 < x2 < ... < xk-1 < xk = b such that (a) f is continuous on (xi-1, xi) for i = 0, 1, 2, ..., k (b) the one sided limits exist as finite numbers Let V be the...

Author: Serge Lang Title: Introduction to Linear Algebra Amazon Link: https://www.amazon.com/dp/3540780602/?tag=pfamazon01-20 Prerequisities: High-School mathematics Level: Undergrad Table of Contents: Vectors Definition of Points in Space Located Vectors Scalar Product The Norm of...

Author: Serge Lang Title: Linear Algebra Amazon Link: https://www.amazon.com/dp/1441930817/?tag=pfamazon01-20 Prerequisities: Some familiarity with matrices and proofs Level: Undergrad Table of Contents: Basic Theory Vectors Definition of points in n-space Located vectors Scalar...

Okay here's my system of equations: x − 3y − 2z = 0 −x + 2y + z = 0 2x + 4y + 6z = 0 Solve the following systems using Gaussian elimination I put it in a matrix and did Gaussian elimination. But I can't find a unique solution and it doesn't end up working out. Is this true? Or am...