Linear transformation Definition and 437 Threads

Homework Statement Define L:R3-->R3 by L(x,y,z)=(y-z,x+z,-x+y). A. Show that L is self-adjoint using the standard orthonormal basis B of R3. B. Diagonalize L and find and orthogonal basis B of R3 of eigenvectors of L and the diagonal matrix. C. Considering only the eigenvalues of L...

[Linear Algebra] Finding T* adjoint of a linear operator Homework Statement Consider P_1{}(R), the vector space of real linear polynomials, with inner product < p(x), q(x) > = \int_0^1 \! p(x)q(x) \, \mathrm{d} x Let T: P_1{}(R) \rightarrow P_1{}(R) be defined by T(p(x)) = p'(x) +...

Homework Statement Prove: Let V be a vector space over the field F . If A,B,C\in L(V) , then A\circ(B+C)=A\circ B+A\circ C . The Attempt at a Solution Note that A\circ B\in L(V) means A\circ B(\mathbf{v})=A(B(\mathbf{v})). Suppose (\alpha_{jk})_{j,k=1}^{n} and...

Given vector spaces V, W over a field, and linear transformation T:V\rightarrow W, prove T(0_{v})=0_{w} where 0_v and 0_w are additive identities of V and W. I'm trying to use the definition of additive identity. So, \forall\vec{v}\in V,\vec{v}+0=\vec{v+0=0} . Where do I go from here?

Homework Statement L:R3->R3 is a dialiation by a factor of 3 of points in the plane W given by the equation z = 0 and a contraction along the line L = span({(1,0,0)}) by a factor of 3. Find [L], but I'm mainly concerned with finding L(e31) Homework Equations z = 0 L(1,0,0) =...

Which of the transformations are onto? 1) T:R2 -> R2, where T(x,y) = (5x-y, 0) I don't know if I'm understanding this correctly but this transformation is NOT onto because if I let 5x-y = a 0 = b this means that b doesn't cover all the range of T? Could someone explain it better if...

T: Mnn => R, where T(A) = tr(A) Attempt: 1) T(kA) = tr(kA) = k tr(A) = k T(A) 2) T(A+B) = tr (A + B) = tr(A) + tr(B) = T(A) + T(B) so it's linear transformation. Am I correct?

I was taught that the columns of a matrix, T, representing a transformation represent the first vector space's basis set and the rows represent the basis set of the range vector space. i.e. T(v_k) = t_1,k*w_1 +... + t_(m,k)*w_m So v_k would be the k-th basis vector of the first space, V...

Homework Statement http://img847.imageshack.us/img847/2783/77597781.jpg Homework Equations See below. The Attempt at a SolutionI get all the way to the last step, but I'm not sure how to perform it.I get to the point where I have: [(T^-1)(ax^2+bx+c)]\alpha= [T^-1]\alpha\beta*[v]\beta= [ a/4...

Homework Statement Prove that T^{2} is a linear transformation if T is linear (from R^{3} to R^{3}. So I understand when a transformation is considered linear, but I don't understand what squaring a transformation does. I don't think it means squaring the result of the transformation but...

Homework Statement I am having lots of trouble understanding how to get the kernel of linear transformations. I get that you basically set it equal to zero and solve. T: P3 → P2 given by T(p(x)) = p΄΄(x) + p΄(x) + p(0) Find ker(T) The Attempt at a Solution So P3 = ax^3 + bx^2 +...

Homework Statement Given: T is a transformation from V -> W and the dim(V) = n and dim(W) = m (I think the dimensions were given for the purposes of another problem) Prove: If T is 1 to 1 and {v1, ..., vk} is a subspace of V that is L.I., then {T(v1), ..., T(vk)} is L.I in W. The Attempt at...

Problem Given a transformation T : P(t) -> (2t + 1)P(t) where P(t) ϵ P3 (a) Show that transformation is linear. (b) Find the image of P(t) = 2 t^2 - 3 t^3 (c) Find the matrix of T relative to the standard basis ε = {1, t, t^2, t^3} (d) Find the matrix of T relative to the basis β1 = {1...

Homework Statement For the linear transformation T: R4 --> R3 defined by TA: v -->Av find a basis for the Kernel of TA and for the Image of of TA where A is 2 4 6 2 1 3 -4 1 4 10 -2 4Homework Equations Let v = a1 b1 c1 a2 b2 c2 a3 b3 c3 a4 b4 c4 The...

Homework Statement V is a vector space consisting all functions f:R->R that is differentiable many times (a) Let T:V->V be the transformation T(f)=f' Find the (real) eigenvectors and eigenvalues of T (b) Let T be transformation T(f)=f" Prove that all real number, m is the eigenvalue of...

Question about linear transformations if you have a matrix such as | 5 6 9 | | 5 0 3 | | 9 -3 -7 | Can it be a matrix transformation? Or does it have to follow the identity matrix? Can be a transformation and the "y" transformation being just makes the it flat on the y axis? or...

Hi all, Here is the problem: If T: V -> W is a linear transformation and S is a linearly dependent subset of V, then prove that T(S) is linearly dependent. Now, I know that the usual proof goes as follows: Since S is linearly dependent, there are distinct vectors v_1, ..., v_n in S and...

Hi, I am trying to prove the following equality Range(T*T) = Range(T*) where T is a linear transformation and * denotes the adjoint. I know I must first show that Range(T*T) Range(T*) and vice versa. so, Let w exist in R(T*T), then there exists a v in vector space V s.t. T*T(v)...

Homework Statement I need to show the following equalities, where T is a linear transformation and * is the adjoint. Null(T*T) = Null(T) Range(T*T) = Range(T) The attempt at a solution I know I have to show that Null(T) is a subset of Null(T*T) and then Null(T*T) is a subset of...

Homework Statement In a given basis \{ e_i \} of a vector space, a linear transformation and a given vector of this vector space are respectively determined by \begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 0\\ 0&0&5\\ \end{pmatrix} and \begin{pmatrix} 1 \\ 2 \\3 \end{pmatrix}. Find the matrix...

Homework Statement How to prove that: the range of a square matrix A (linear transformation) to the power of n+1 is a subspace of the Range of A to the power n, for all n >= 1? i.e. Range (A^(n+1)) is a subspace of Range (A^n) Homework Equations The Attempt at a Solution I...

Let T:P2\rightarrowP2 be given by T(x-1)=1-x T(x2-2x)=-1+x-2x2 T(3-x2)=-1+2x+3x2 Find the matrix for T with respect to the standard basis B={1,x,x2}for P2 To be honest, I have no idea where to start. Help would be greatly appreciated

Homework Statement Prove: If V and W are finite-dimensional vector spaces such that dim(W)<dim(V), then there is no one-to-one linear transformation T:V-->W The Attempt at a Solution I don't know how to do a well thought out proof.

Determine whether the linear transformation T is one-to-one a) T:P2 --> P3, where T(a+a1x+a2x^2)=x(a+a1x+a2x^2) b) T:P2 --> P2, where T(p(x))=p(x+1) I'm having difficulty because my teacher never showed examples like this one. Please help me on the procedure and solution.

I have a matrix of the form X = [A B], where A and B are matrices of equal dimensions (M x N). I am looking for an elegant transformation to obtain Y = [A; B]. That is, the blocks are now stacked vertically. Normally, I'd look for a solution of the form Y = VXW, where V is (2M x M) and W is...

I am a graduate assistant and was asked a question about FLTs (Mobius Transformations). The student was asked to prove that any FLT can be written as an FLT with determinant 1. However, I can't make sense of that. If I look at the possible Jordan Canonical forms of 2-by-2's, it would seem...

Homework Statement Find a linear transofmration from X={x1,x2,x3} to U={u1,u2,u3} which will remove the cross product term in the quadratic form of equation 2X12+4X22+5X32-4X1X3 and thus write the resulting quadratic form in u1,u2,u3. Homework Equations The Attempt at a Solution No...

Homework Statement Give an example of a linear transformation T: R2 -> R2 such that the null space is equal to the range. Homework Equations null space and range The Attempt at a Solution I have been trying to come up with a solution but I cannot figure it out. What might be a...

Hello everyone, I am a year 10 student. I am working on Transformation and there are 2 problems in the book that I actually have no idea to solve. Unfortunately, I am going to have a sudden exam in the next three days. Hope that somebody can help me, thank you a lot! Or If you have any books...

Homework Statement State why it is that if V is finite-dimensional, and if S and T in HomV satisfy S\circ T = I, then T is invertible and S=T^-1 2. The attempt at a solution Isn't this obvious? I don't understand why any elaboration is necessary

Hi, I've come across this result which says that if there are two isomorphic vector spaces with a transformation between them, then that transformation must be linear. Can anyone help me prove this? For instance, if I have a transformation T: Z -> Z where Z is the set of integers, T(z) =...

Homework Statement Find the standard matrix for the linear transformation T: R^3-->R^3 satisfying: T([1 2 2]) = [1 0 -1], T([-1 -4 -5]) = [0 1 1], T([1 5 7]) = [0 2 0] All of the vectors are columns not rows, I couldn't type them correctly as columns. The Attempt at a Solution I...

Given a linear tansformation T of a vector space V (over a field K) with eigenbasis {v_{1},...,v_{n}}, and a (non-trivial) subspace W of V such that T(W) is a subset of W, a lecturer keeps using the result that W will contain an eignvector for T. I can see why this would be the case if the field...

Homework Statement Let T: V --> W be a linear transformation and let U be a subspace of W. Prove that the set T-1 (U) = {v E V: T (v) E U} is a subspace of V. What is T-1 if U = {0}? Homework Equations The Attempt at a Solution Since U is a subspace, k(v) = ku. Also, if u and v...

Homework Statement I know I am suppose to show that the matrix is closed under addition and multiplication properties, but is it POSSIBLE for me to show that the vector columns can be spanned in the given R^m (assume that the Linear Transformation happens from R^n -> R^m) ? For example, [x...

Nevermind -- Polygons and Polywags.

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 I'm trying to prove the following: Let V be a finite dimensional vector space and let T be a linear operator on V. Suppose that the rank of T is equivalent to the rank of T^2. Then the range and the nullspace of T are disjoint. 2. The attempt at a solution I've played...

1) Let L:R3 >>>R3 be defined by L([1 0 0]) = [1 2 3], L([0 1 0]) = [0 1 1], L([0 0 1]) = [1 1 0] How to prove that L is invertible? I have the idea of one-to-one and onto, but I do not know how to apply them to this proof. 2) Find a linear transformation L:R2 >>>R3 such that {[1 -1 2], [3 1...

Homework Statement Will the values given below define a unique linear transformation? If so, find the value of T for an arb. domain element. T(3,5) = (1,2) and T(2,3) = (6,7) Homework Equations The Attempt at a SolutionT(a[3,5]) + T(b[2,3]) = aT[3,5] + bT[2,3]...

Let L:p2 >>> p3 be the linear transformation defined by L(p(t)) = t^2 p'(t). (a) Find a basis for and the dimension of ker(L). (b) Find a basis for and the dimension of range(L). The hint that I get is to begin by finding an explicit formula for L by determining L(at^2 + bt + c). Does...

I can't figure out how to take the first bite out of this one. Homework Statement Let T1: R^2 --> R^2 and T2: R^2 --> R^2 have the indicated properties. Find matrices A, B, and C such that: T2T1x=Ax, T1T2x=Bx, (T1+T2)x=Cx Homework Equations T1e1=(1,3), T1e2=(2,2), T2e1=(-1,1)...

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...

Hi, the question is for the transformation : T: M2,3 -> P2 T ( A11 A12 A13) = (A11 + A13)x^2 + (A21 - A22)x + A23 ( A21 A22 A23) Are the following in the linear transformation? W1=x^2 + 2x + 1 W2=x-2 Attempt: I figured that w would be in image if there exists.. (A11 + A13)x^2 + (A21 -...

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 Let f:R[X] -> R[X] be the linear transformation sending a polynomial P(X) to f(P(X))= P(X+1) - P(X). a) Let f4: R4[X] -> R[X] be the linear transformation induced by restriction of f to the R-vector space of polynomial of degree at most 4. Determine the kernel and the...

Homework Statement My buddy was asking me this question. It's from his linear algebra homework. F: R2 --> R3 F[(x1 x2)] = (x1 x2 0) Homework Equations I can't remember the definition of "linear transformation." Hopefully it's not too complicated. The Attempt at a Solution I don't think...

Homework Statement Let T be an invertible linear transformation from R2 to R2. Let P be a parallelogram in R2. Is the image of P a parallelogram as well? Explain. P is a parallelogram in the first quadrant without any specified points. Homework Equations The Attempt at a...

Homework Statement Let U be a subspace of V. Suppose that U is a invariant subspace of V for every linear transformation from V to V. Show that U=V. Homework Equations no The Attempt at a Solution Assume U is not trivial: Now we only need to show that U = V. Let dimV = n: We can...

Every definition I find on Google utilizes vector spaces and mapping which are mathematical. If you have to use vector spaces and mapping please explain the mathematics behind it. Thanks.