Linear algebra Definition and 999 Threads

this is for a first course in LA,can I ask the avid readers :Which teaches more of the theory and meanings Please?Which is better?: Introduction to Linear algebra and linear algebra by serge lang? Linear algebra by Shilov? Halmos P. R Linear Algebra Problem Book ?

Homework Statement Here is the problem: http://img801.imageshack.us/img801/6770/oaza.png Homework Equations None really, just gauss jordon elimination I assume unless I am missing out on something The Attempt at a Solution First I multiplied the first row by -5 then added...

Homework Statement R(M) and C(M) are the row and column spaces of M. Let A be an nxp matrix, and B be a bxq matrix. Show that C(AB) = C(A) when the orthogonal complement of R(A) + C(B) = R^p (i.e. the orthogonal complement of R(A) and C(B) span R^p). Homework Equations I know that the...

I'm going to do linear algebra for first time,so I look online and found Halmos' Linear Algebra Problem Book https://www.amazon.com/dp/0883853221/?tag=pfamazon01-20 ,does it fit in the course(does it teach everything without much prequistes?),what book do I use after that? Any suggestion or...

Homework Statement For fixed m ≥ 1, let ##\epsilon(i,j)## denote the m x m matrix ##\epsilon(i,j)_{rs} = \delta_{ir}\delta_{js}##, where i,j may denote any integers in the range 1 ≤ i,j ≤ m. (a) When m = 4, write out all ##\epsilon(i,j)## explicitly and label them correctly. The attempt at a...

Homework Statement Let K be the closure of Qu{i}, that is, K is the set of all numbers that can be obtained by (repeatedly) adding and multiplying rational numbers and i, where i is the complex square root of 1. Show that K is a Field. Homework Equations The Attempt at a...

Homework Statement Let V1 and V2 be vector spaces over the same field F. Let V = V1 X V2 = {f(v1, v2) : v1 \in V1; v2 \in V2}, and define addition and scalar multiplication as follows.  For (v1, v2) and (u1, u2) elements of V , define (v1, v2) + (u1, u2) = (v1 + u1, v2 + u2).  For...

Hello PF! Homework Statement Prove the following: if u and v are two vectors in Rn such that u\cdotw = v\cdotw for all wεRn , then we have u = v Homework Equations The Attempt at a Solution u\cdotw - v\cdotw = 0 w\cdot(u - v) = 0 I'm not sure what to do after applying the...

Homework Statement Let A be a nxn matrix, and I the corresponding identity matrix, both in the real numbers ℝ. Assume that A^m=0 for a positive integer m. Show that I-A is an invertible matrix. Homework Equations The Attempt at a Solution

Hello MHB, I am stuck at this problem for quite a long time now. Problem. Let $F_p$ denote the field of $p$ elements, where $p$ is prime. Let $n$ be a positive integer. Let $V$ be the vector space $(F_p)^n$ over the field $F_p$. Let $GL_n(F_p)$ denote the set of all the invertible linear...

Hi all, The title is pretty much the question. My friend (who wants to go to graduate school in Physics) is between two courses the math department offers: an upper-level linear algebra course, and a second course in ODEs. Here are the course descriptions: and The school in question is...

Can you recommend one these book as a first andd concise introduction to linear algebra ,which is better?: Linear Algebra (Dover Books on Mathematics) Georgi E. Shilov https://www.amazon.com/dp/048663518X/?tag=pfamazon01-20 Linear Algebra: A Modern Introduction David Poole...

The matrix is an example of a Linear Transformation, because it takes one vector and turns it into another in a "linear" way. Hi could you explain it what exactly the bold part suggest? What is "another in a "linear" way"?

Sorry if this is the wrong place to ask this, but I think linear algebra is the best place to ask my question. Feel free to move this thread elsewhere if I am wrong. I would like to know how I can use linear algebra to help me figure out when I am deriving an equation if the derivation I...

In my linear algebra class we previously studied how to find a basis and I had no issues with that. Now we are studying the basis of a row space and basis of a column space and I'm struggling to understand the methods being used in the textbook. The textbook uses different methods to find these...

Hi all, Can linear algebra used to deal with non linear systems? and why linear algebra is 'linear'? :( -Devanand T

Homework Statement Let L: V →V be a linear mapping such that L^2+2L+I=0, show that L is invertible (I is the identity mapping) I have no idea how to solve this problem or how to start,I mean this problem is different from the ones I solved before, the answer is "The inverse of L is -L-2 "...

Homework Statement Find the value of k such that no solutions are shared between the planes x + 2y + kz = 6 and 3x + 6y + 8z = 4Homework Equations The Attempt at a Solution I figured that if I calculated \vec{n_1} \times \vec{n_2} = \vec{0}, and then take the magnitude of this vector...

Homework Statement Find a system of two equations in two variables, x and y, that has the solution set given by the parametric representation x=t and y=3t-4, where t is any real number. Homework Equations x=t and y=3t-4, where t is any real number The Attempt at a Solution y=3x-4...

Homework Statement The part I'm having problems with is where the last two expressions in 4.13 are equated. Why is xtKx*equal to x*^tKx*? The Attempt at a Solution xtKx* is an inner product and due to symmetry is equal to x*^tKx, but wouldn't equating x to x* mean every <x,y> = <x,x> =...

Homework Statement Prove that if a∈F (where F represents ℝ or ℂ), v∈V (where V is a vector space) and av = 0, then a= 0 or v = 0. Homework Equations The axioms for a vector space may be relevant. The Attempt at a Solution Case 1 (v = 0): Suppose that a∈F, v∈V, and av = 0. Also, let u∈F...

Homework Statement In ##E^3##, given the orthonormal basis B, made of the following vectors ## v_1=\frac{1}{\sqrt{2}}(1,1,0); v_2=\frac{1}{\sqrt{2}}(1,-1,0); v_3=(0,0,1)## and the endomorphism ##\phi : E^3 \to E^3## such that ##M^{B,B}_{\phi}##=A where (1 0 0) (0 2 0) = A (0 0 0)...

Hi, I've searched this forum and narrowed my choices down to these three books: linear algebra - friedberg linear algebra - hoffman &kunze linear algebra - serge langCould anyone please compare these three books? I'm a high school student and haven't studied linear algebra before. (I can...

Why is it that for arbitrary z, A^{T}z = 0 and b^{T}z ≠ 0 when there does not exist an x such that Ax = b, i.e. that b is not in the range space of A, where A is an n x m matrix?

This is something that I haven't really found much info on. I'm a student attending McGill university, and I have a choice between taking Honours Applied Linear Algebra and Honours Algebra. Linear algebra: Mathematics & Statistics (Sci) : Matrix algebra, determinants, systems of linear...

So I wanted to begin studying linear algebra prior to the start of my first academic year at university. One textbook I want to use is "Linear Algebra Done Right" by sheldon axler. Any feedback on this text would be nice.

I just had my last Linear Algebra class, and I didn't get a chance to ask the one question that has been bugging me ever since we started in earnest with matrices. Why aren't there cube matrices? I mean, mathematical entities where numbers are 'laid out' in 3d not in 2d (not quite...

Homework Statement ##\phi## is an endomorphism in ##\mathbb{E}^3## associated to the matrix (1 0 0) (0 2 0) =##M_{\phi}^{B,B}##= (0 0 3) where B is the basis: B=((1,1,0),(1,-1,0),(0,0,-1)) Find an orthonormal basis "C" in ##\mathbb{E}^3## formed by eigenvectors of ##\phi## The...

Homework Statement I've tried to solve the following exercise, but I don't have the solutions and I'm a bit uncertain about result. Could someone please tell if it's correct? Given the endomorphism ##\phi## in ##\mathbb{E}^4## such that: ##\phi(x,y,z,t)=(x+y+t,x+2y,z,x+z+2t)## find: A) ##...

Homework Statement I have a quick question about the proof below. Let A be an nxn matrix. Prove that A is singular if and only if λ=0 I searched the proof online, and they did it using Ax=0 However, When I tried doing on my own my solution was this If A is singular then the...

Homework Statement Evaluate the laplace transform of {t2e7tsinh(3t)} Homework Equations Laplace transform of {tnf(t)}=(-1)ndn/ds2 * F(s) The Attempt at a Solution I've replaced it with (-1)2d2L{e7tsinh(3t)} I'm not sure how to proceed, though, as I don't really see how to take...

Homework Statement Write a selfadjoint endomorphism ## f : E^3 → E^3## such that ##ker(f ) = L((1, 2, 1)) ## and ## λ_1 = 1, λ_2 = 2## are eigenvalues of f The Attempt at a Solution I know ##λ_3=0## because ́##ker(f ) ≠ {(0, 0, 0)}## and ## (ker(f ))^⊥ = (V0 )^⊥ = V1 ⊕ V2 ## due to...

I've had a proof based linear algebra course as a freshman, where I learned that the spectrum of an operator was the set of the eigenvalues of that operator. Now in quantum mechanics I learned that this isn't true and that the spectrum of an operator can contain infinitely more numbers...

Hello All, First post here. I going to be taking summer school to get some classes out of the way and am deciding whether or not to take Ordinary Diff. Eqn's or Linear Algebra. Any help would be appreciated on the difficulty of the courses and what is covered. Thanks in advance.

1. Find A, a 2x2 matrix, where A^{1001}=I_{2}2. I know that that if A^{2}=I_{2}, then A is either a reflection or a rotation by π. 3. If I use advantage of that fact that A in A^{2}=I_{2} is a rotation by π then I know that A^{1001}=I_{2} is true when A is a rotation by 2π/1001 Is there...

Homework Statement Consider the following system: x + by = -1 ax + 2y = 5 Find the conditions on a and b such that the system has no solution, one solution or infinitely many solutions. Homework Equations General Algebra really. The Attempt at a Solution Previously we had been...

So we are given two equations: $$ \ddot{x} - \dot{x} - x = cost (t) $$ and $$ x(t) = a sin(t) + b cos(t) $$ The question asks to find a and b. How would one go about doing this? I thought maybe substituting the $$ cos(t) $$ from equation 1 into equation 2 would work but then what...

I'm looking for a linear algebra textbook slightly above beginner level to get a good grasp of certain topics. Specifically, I want to understand exactly what the determinant of a matrix is and its relation with volume, linear dependence/independence, etc I recently figured out how to get...

Homework Statement Hello, I took my quiz today, and had to find a basis for an orthogonal compliment, would it be incorrect to not factor out the alphas and betas? Homework Equations The Attempt at a Solution

Hello guys, I will be taking Linear Algebra (Intro.) and Discrete Math in the Fall. I heard that these two courses are different from the Calculus sequence. I am afraid since I am not good with proofs. Will I be able to do well as long as I put in the time? Can you guys give me an advice so I...

Homework Statement Determine whether W is a subspace of the vector space: W={(x,y):y=ax, a is an integer} , V=R^2 Homework Equations noneThe Attempt at a Solution Is u+v in W? Let u = (u,au) and v = (v,av) u+v = (u,au) + (v,av) = (u+v, au + av) = (u+v, a(u+v)) If x = u+v => u + v = (x,ax) =>...

Homework Statement Let x and y be linearly independent vectors in R2. If ||x||=2 and ||y||=3, what if anything can we conclude about the possible values of |xTy| I know that ||x||*||y||cos(∅)=|xTy| If ||x||=2 and ||y||=3, then we can get either 6 or -6, I'm not sure if this is...

Homework Statement Find the matrix for the transformation which first reflects across the main diagnonal, then projects onto the line 2y+√3x=0, and then reflects about the line √3y=2x Homework Equations Reflection about the line y=x: T(x,y)=(y,x) Orthogonal projection on the x-axis...

Homework Statement Let M2 denote the set of all 2x2 matrices. We define addition with the standard addition of matrices, but with scalar multiplication given by: k \otimes [a b c d] = [ka b c kd] (note that they are matrices) Where k is a scalar. Which of the...

Homework Statement Let S be a subset of R^{4}: S = {v_{1},v_{2}, v_{3}, v_{4}} = { [1,3,2,0] [-2,0,6,7] [0,6,10,7] [2,10,-3,1] } Determine whether S spans R^{4}. Homework Equations span(S) = {V| V = av_{1}+bv_{2}+cv_{3}+dv_{4}} The Attempt at a Solution When I row...

Homework Statement Let T: R3--> R4 be a linear transformation. Assume that T(1,-2,3) = (1,2,3,4), T(2,1,-1)=(1,0,-1,0) Which of the following is T(-8,1-3)? A. (-5,-4,-3,-8) B. (-5,-4,-3,8) C. (-5,-4,3,-8). D.(-5,4,3,-8) E (-5,4,-3,8) F. None of the above.Homework Equations I really have no...

Homework Statement Find all subspaces of the vector spaces: (ℝ+,.) , (ℝ^2 +,.) , (ℝ^3 +,.) The Attempt at a Solution For ℝ the only subspace i can think of is {0} For ℝ^2 if found {0} R^2 itself and any set of the form L=cu for u≠0. For ℝ^3 if found those of R^2 plus R^3 . Are...

Homework Statement 1) The set H of all polynomials p(x) = a+x^3, with a in R, is a subspace of the vector space P sub6 of all polynomials of degree at most 6. True or False? 2) The set H of all polynomials p(x) = a+bx^3, with a,b in R, is a subspace of the vector space P sub6 of all...

Homework Statement For each of the following matrices, determine a basis for each of the subspaces N(A) A=[3 4] [ 6 8]Homework Equations The Attempt at a SolutionSo reducing it I got [1 4/3] [0 0] I know x2 is a free variable I set x2 = to β and found...

Homework Statement let b1=(1,1,0)T ;b2=(1 0 1)T; b3=(0 1 1)T and let L be the linear transformation from R2 into R3 defined by L(x)=x1b1+x2b2+(x1+x2)b3 Find the matrix A representing L with respect to the bases (e1,e2) and (b1,b2,b3) Homework Equations The Attempt at a Solution First...