Basis Definition and 1000 Threads

Homework Statement Find a basis of the linear space and thus determine the dimension. q(t) : q'(1)=q(2) Where q is a subset of Q_2: all polynomials of degree less than or equal to 2. I had to change it from p to q so the forum didn't make it a smiley... The Attempt at a Solution...

Homework Statement Determine the Jordan Normal form and find some Jordan basis of the matrix 3 -3 1 A = 2 -2 1 2 -3 2 Homework Equations dim(A) = rk(A) + dimKer(A) The Attempt at a Solution My problem here is that my lecturer seems to be doing...

Homework Statement From Calculus we know that, for any polynomial function f : R -> R of degree <= n, the function I(f) : R -> R, s -> ∫0s f(u) du, is a polynomial function of degree <= n + 1. Show that the map I : Pn -> Pn+1; f -> I(f), is an injective linear transformation, determine...

Homework Statement (X,<,>) is a inner product space over R {ei}i in N is an orthonormal set in X Show that if every element u in X can be written as a linear combination u = \sum_{i=1}^\infty a_i e_i then {ei}i in N is a basis for X Homework Equations Let {ei}be a sequence of...

Homework Statement A voltage of a car battery is 12V, the current requires for movement is 200A. When we turn the lights on the voltage drops to 11.1V. The power of each backlight is 55 watt, the power of each front light is 12 watt. A) Calculate the internal resistance of the battery...

Homework Statement Suppose the matrix standard matrix for a linear trnaformation T: R^2 --->R^2 is[PLAIN]http://www.texify.com/img/%5CLARGE%5C%21%5Cbegin%7Bequation%7D%5Cbegin%7Bpmatrix%7D2%20%26%20-3%20%5C%5C%200%20%26%201%5Cend%7Bpmatrix%7D%5Cend%7Bequation%7D.gif Find the matrix T with...

Homework Statement B1 = {[1,2], [2,1]} is a basis for R2 B2 = {[1,-1], [3,2]} is a basis for R2 Find the change of basis matrix from B1 to B2 Homework Equations [B2 | B1] The Attempt at a Solution For some reason I can not solve this. I keep ending up with the matrix...

Looking at Lagrangians and field equations for different spin all the derivations seem to lack a common basis; they appear to lack any deep relation. Is there a unique way to understand the different forms like Klein-Gordon, Dirac, Maxwell (Yang-Mills), etc. from a common basis which is valid...

Homework Statement A lot of my homework asks me to determine if a given matrix (sequence of vectors) is a basis or not. Homework Equations The Attempt at a Solution Can I just find the reduced echelon form of a given matrix and see if it is linear independent or linear...

Homework Statement Homework Equations In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system".[1] In more general...

My apologies about lack of precision in nomenclature. So I wanted to know how to express a certain idea about choice of basis on a manifold... Let's suppose I am solving a reaction-diffusion equation with finite elements. If I consider a surface that is constrained to lie in a flat plane or...

How do you extend a vector (let's use vector (1,2,3) for example) to an orthogonal basis for R^3?

Homework Statement Let Y:P3(R) onto P2(R) is defined by T(a0+a1z+a2z2+a3z3)=a1+a2z+a3z2. Find bases for null (T) and range (T). What are their dimensions? Homework Equations The Attempt at a Solution Well, I am assuming the range is the a1+a2z+a3z^2, so it has dimension of 3, and...

Density matrix and von Neumann entropy -- why does basis matter? I'm very confused by why I'm unable to correctly compute the von Neumann entropy S = - \mathrm{Tr}(\rho \log_2{\rho}) for the pure state | \psi \rangle = \left(|0\rangle + |1\rangle\right)/ \sqrt 2 Now, clearly the simplest...

Please look at the link: http://gyazo.com/5f57caddf7dc76aa61d387f2915d29fe.png

Homework Statement Find an orthonormal basis for the subspace of R^4 that is spanned by the vectors: (1,0,1,0), (1,1,1,0), (1,-1,0,1), (3,4,4,-1) The Attempt at a Solution When I try to use the Gram-Schmidt process, I am getting (before normalization): (1,0,1,0), (0,1,0,0), (1,0,-1,2)...

"Usefulness" of Basis for a Vector Space, General? Hi, Everyone: I am teaching an intro class in Linear Algebra. During the section on "Basis and Dimension" a student asked me what was the use or purpose of a basis for a vector space V. All I could think of is that bases allow us to...

Homework Statement The question states: Consider the subset S of R^4 given by: S={(2,3,-1,7), (1,0,1,3), (0,3,-3,1), (12,15,-3,29)} i) Decide whether the vectors in S form a linearly independent set. ii) Let V be the vector subspace of R^4 spanned by the vectors of S, i.e: V=span{...

1) Let S be an ordered basis for n-dimensional vector space V. Show that if {w1, w2, ..., wk} is a linearly independent set of vectors in V, then {[w1]s, [w2]s,...,[wk]s} is a linearly independent set of vectors in R^n. What I got so far is w1 = a1V1 + a2V2 + ... + anVn so, [w1]s = [a1...

1.Find the degree and basis for Q(3^1/2,7^1/2) over Q. 2.For any positive integers a, b, show that Q(a^1/2+b^1/2)=Q(a^1/2,b^1/2) Ideas: 1. Well I know if I looked at (3)^1/2 over Q Then (3)^1/2 has minimal polynomial x^2-3, so degree 2 over Q (7)^1/2 has minimal polynomial x^2-7 so...

Homework Statement Find a basis for the subspace of R4 spanned by S. Homework Equations S: {(2,9,-2,53), (-3,2,3,-2), (8,-3,-8,17), (0,-3,0,15)} I've attempted this using a matrix and row reducing it. I'm just not sure if there's another simpler way, as I keep on getting incorrect...

Homework Statement Let Ψ: Mat2x2(R) -> Mat2x2(R) be defined as: [a,b;c,d] -> [a+b, a-c; a+c, b-c] Find a basis for the image of Ψ. Homework Equations None, AFAIK. The Attempt at a Solution I started by using the standard basis, B, for Mat2x2(R) to get [u]B [with u in Mat2x2(R)] as...

Homework Statement Hi fellows, If we are given 3 vectors (e.g X1, X2, X3) in R^4, how would we find X4 such that {X1, X2, X3, X4} is a linearly independent set? Homework Equations The Attempt at a Solution I tried something like this: aX1 + bX2 + cX3 + dX4 =0, but it didn't...

Homework Statement Let c be a real scalar not equal to zero. Prove that if a set S ={v1, v2, ... , vn} is a basis for V, then set S1= {cv1, cv2, ... , cvn} is also a basis of V. Homework Equations A set is a basis if it spans a subspace and is not collinear. The Attempt at a...

Equation of a Surface relative to the standard basis is X1^1 + 2X2^2 + 3x^3 -4x1x2 -4x2x3 =0 Now the question ask to find the equation of above surface relative to the cordinate system with basis vectors F1= (2/3,2/3,1/3) F2= (1/3,-2/3,2/3) F3= (2/3,-1/3,-2/3) Now i found the transition...

Hi All, I recently came across the interesting notion of constructing the minimal set of nxn matrices that can be used as a basis to generate all nxn matrices given that matrix multiplication, and addition and multiplication by scalar are allowed. Is there a way to construct an explicit set...

Hi, Everyone: I have a quadratic form q, defined on Z⁴ , and I know the value of q on each of the four basis vectors ( I know q is not linear, and there is a sort of "correction" for non-bilinearity between basis elements , whose values --on all pairs (a,b) of...

hi.. can anyone say what is the concept behind change of basis.. y do we change a vector of one basis to another?

Hello, I am trying to express a given wavefunction through different basis, momentum and position. Look at 5.2(b) and (c) through the link: http://qis.ucalgary.ca/quantech/443/2011/homework_five.pdf" I complete part (b) by doing the following...

Homework Statement Suppose L:R^2 -> R^3 Find the matrix representing L(x) = Tx with respect to the ordered basis [u1,u2] and [b1.b2,b3] Homework Equations The Attempt at a Solution I've excluded the actual values since i can do the computation. Just wanted to make sure these steps are ok and...

Homework Statement Whenever LA talks about ℝn, do they mean just the n? Ex. Let's say I have two vectors \begin{bmatrix}1\\ 0\\ 0\end{bmatrix}\begin{bmatrix}0\\ 1\\ 0\end{bmatrix} Now it is tempting to say that the two vector is a basis for ℝ2. Now my professor tells me that it isn't a...

Homework Statement Let S be a subspace of a vector space V. Let B be a basis for V. Is there a basis C for S such that C \subseteq B? not really sure how to approach this... any hints?

Homework Statement Let V be the subspace spanned by the following vectors: [ 0]...[ 1 ]...[2] [ 2]...[ 1 ]...[5] [-1]...[3/4]...[0] Determine a basis for V. The Attempt at a Solution I'm not quite sure how to start here. Would placing the vectors in a matrix and deriving its...

Someone posted this on another forum, and, not knowing enough about it to supply a satisfying answer, I figured I'd ask you guys.

Homework Statement If possible, find a basis a = {a1, a2, a3} of P2(R) such that... [2 + 5x + 4x^2]a = [1, 2, 3], [1 + x + x^2]a = [4,1,2] and [x + x^2]a = [3, -5, 1] 2. The attempt at a solution Basically, we have something like Ax = b for each of these, right? A* [2,5,4] =...

Hi, Just curious as to what is the basis of the margin of error given in polls, e.g., in statements of the form:" 30% of people are in favor of candidate x. The poll has a margin of error of +/- 5 %" Thanks.

Hello, I am considering the set of all (differentiable) even functions with the following properties: 1) f(x)=f(-x) 2) f(0)=a_0, with a_0\in \mathbb{R} 3) f(n)=0, where n\in \mathbb{Z}-\{0\} One example of such a function is the sinc function sin(\pi x) / \pi x. Is it possible to find...

Homework Statement Hi, i am applying the gram-schmidt procedure to a basis of {1,2x,3x^2} with inner product <p,q> = \int p(x)q(x) from 0 to 1. i am unsure what to do with the inner product Homework Equations The Attempt at a Solution I have followed the procedure i have for...

Homework Statement Note: the vectors are column vectors, not row vectors. Latex is not working for me right now. Find an orthonormal basis u1, u2, u3 of R3 such that span(u1) = span [1 2 3] and span(u1,u2) = span { [1 2 3], [1 1 -1] } Homework Equations The Attempt at...

Prove that t2+2t+1,t2+t, t2+1 is a basis for the space of quadratic polynomials P(2).

Homework Statement The last 2 parts of the attached photo. (4 and 6 marks) Im really not sure how to go about them in a (clever) way that won't take 2 hours. Homework Equations Possibly the fact that the product of the raising/lowering operators, J-J+ = J2x + J2y Answers to previous...

Homework Statement Let t \in L(V,W). Prove that t is an isomorphism iff it carries a basis for V to a basis for W.Homework Equations L(V,W) is the set of all linear transformations from V to WThe Attempt at a Solution So I figured I would assume I have a transformation from a basis for V to a...

a)Find a basis for the space of 2x2 symmetric matrices. Prove that your answer is indeed a basis. b)Find the dimension of the space of n x n symmetric matrices. Justify your answer.

Homework Statement find the basis of the nullspace of this matrix \begin{pmatrix} 1&1&1&-1 \\ 0&0&1&3 \end{pmatrix}Homework Equations The Attempt at a Solution i forget it. i first substitue 0 and 1 for last row but what about the first row? Substiute 0 and 1 again and this will give 4 basis...

Homework Statement Find a basis for each of the spaces and determine its dimension: The space of all matrices A=[a b, c d] (2x2 matrix) in R^(2x2) such that a+d=0 Homework Equations The Attempt at a Solution So I jumped at this question without knowing too much about spaces and...

Homework Statement I've been browsing the Internet but can't find a straightforward explanation for a procedure on how to find the image and kernel of a matrix. Question: Find a basis of the image of A, and a basis of the kernel of A. \[ A = \left[ {\begin{array}{ccc} 1 & 2 & 1 \\...

Homework Statement The mass of a van with a driver is 2000 kg . When the van accelerates, the velocity increases with a uniform acceleration of 3.0 m/s2. Homework Equations a) The van starts at rest. Find the velocity after 4.0 s. b) How far does the van travel in the first 4.0 s? c)...

Making a change of basis in the matrix representation of a linear operator will not change the eigenvalues of that linear operator, but could making such a change of basis affect the geometric multiplicities of those eigenvalues? I'm thinking that the answer is "no", it cannot.. Since if...

Let X denote the set of real symmetrical 3X3 matrix. Then (X,R) forms a linear space. What will be a basis set for this linear space? I would appreciate if someone can help me with the question. My understanding is in R3 space there could be many 3X3 matrix that could be the basis set for...

(a) Let A (matrix) =c1= [1,2,1], c2 = [0,1,2], c3 = [3,-2,-1] be a matrix (c1,c2,c3 refer to the columns of the matrix A, which is a 3x3 matrix) expressed in the standard basis and let w1 = (0,0,1)T, w2 = (0,1,2)T , w3 =(3,0,2)T , find the vector AUE in w basis. (b). Referring to problem (a)...