Linear transformations Definition and 193 Threads

Homework Statement Given A = \left(\begin{array}{ccc}1&-1&1\\0&1&1\end{array}\right) Why isn't Latex working for above array :( Define a transformation as T: \Re^{n} -> \Re^{m} T(\vec{x}) = A \vec{x} 1) a. What is n? b. What is m? 2) Find \vec{x} , if possible, given that...

Homework Statement Let V be an n-dimensional vector space over R, and let S and T be linear transformations from V to V. (i) Show that im(S+T) \subseteq im(S) + im(T) (ii) Show that r(ST) \leq min(r(S),r(T)), and that n(ST) \leq n(S) + n(T) Homework Equations none that i can think...

Homework Statement if Sa: R2 -> R2 is a rotation by angle a counter-clockwise if Tb: R2 -> R2 is a reflection in the line that has angle b with + x-axis Are the below compositions rotations or reflections and what is the angle? a) Sa ○ Tb b) Ta ○ Tb Homework Equations I don't...

Homework Statement If f,g are Riemann integrable on [a,b], then for c,d real numbers, (let I denote the integral from a to b) I (cf + dg) = c I (f) + d I (g) Homework Equations The Attempt at a Solution I have the proofs for c I(f) = I (cf) and I (f+g) = I (f)...

I wasn't sure if this counted as intro physics. Feel free to move if I have it in the wrong place. Homework Statement In class we learned some linear transformations where we have a stationary observer and another moving near the speed of light. Describing the reference frames: s' -> x'=u't'...

Homework Statement Not really a problem per se; more of an issue with some aspects of linear transformations. We've learned that a linear combination of linear transformations is defined as follows: (c_1T_1+c_2T_2)(\vec{x})=c_1T_1(\vec{x})+c_2T_2(\vec{x})\,\,\,\,\,\, \vec{x}\varepsilon...

1. R\circF\circR-1=S where F denotes the reflection in the x-axis where S is the reflection in the line y=x where R = R\pi/4 : R2 \rightarrow R2 3. An attempt I have found that the standard matrix for R = [cos\theta sin\theta]...

Homework Statement Let T \in L(V, W), where dim(V) = m and dim(W) = n. Let {v1, ..., vm} be a basis of V and {w1, ..., wn} a basis for W. Define the matrix A of T with respect to the pair of bases {vi} and {wj} to be the n-by-m matrix A = (aij), where T(v_{i}) =...

Homework Statement Let A: E \rightarrow F be a linear transformation between vector spaces (of any dimension) and let X be a subset of F with the following property (which is only a conditional): IF X \subseteq Im(A) THEN A is surjective. ... (*) Prove that X is a generating set for...

Homework Statement Suppose that A is a real symmetric n × n matrix. Show that if V is a subspace of R^n and that A(V) is contained in V , then A(V perp) is contained in V perp. Homework Equations A = A_T (A is equal to its transpose) The Attempt at a Solution I have no idea...

Homework Statement [PLAIN]http://img219.imageshack.us/img219/2950/linl.jpg Homework Equations The Attempt at a Solution Is this how I do part (iii)? From (ii) I get: M^{\mathcal C}_{\mathcal C} (\phi) = \begin{bmatrix} 1 & 3 & 2 \\ 1 & -3 & 0 \\ 0 & 0 & 2 \end{bmatrix}...

Homework Statement Prove that if V is a finite-dimensional vector space, then the space of all linear transformations on V is finite-dimensional, and find its dimension. Homework Equations The Attempt at a Solution

I am having trouble with this problem: Let T:V->W be a linear transformation. Prove that T is one-to-one if and only if dimension of V = dim(RangeT). I know that in order to be a linear transformation: 1) T(vector u + vector v) = T(vector u) + T(vector v) and 2) T(c*vector u) =...

Homework Statement Write down 3x3 matrices A, B, C such that when the vectors in R2 are expressed in homogeneous coordinates, the product ABC first translates vectors by (-1, 2), then reflects them about the line y=-x and finally scales them by 2. using your matrix ABC, determine the image...

I don't quite understand the idea that (as my book says) every linear transformation with domain Rn and codomain Rm is a matrix transofrmation... I mean i get the idea of what a linear transformation is (sorta like a function) but it gives the theorem: Let T: Rn -> Rm be linear. Then there is...

Homework Statement The Attempt at a Solution I think I first need to find T(e2)=? and T(e2)=? and then combine those into a matrix. I am having trouble starting to solve for T(e1) and T(e2) so far I have [1] = alpha [1] + beta [3] [0] [2]...

Homework Statement which of the following are linear transformations. a) L(x,y,z) = (0,0) b) L(x,y,z) = (1 ,2, -1) c) L(x,y,z) = (x^2 + y, y - z) The Attempt at a Solution I know that L is a linear transformation if L(u + v) = L(u) + L(v) and L(ku) = kL(u). I am not sure how to...

For the linear transformation, T: R^2\rightarrow R^2, T(x,y) = (x^2,y) find the preimage of.. f(x)= 2x+1 I have no trouble with these types of problems when it comes to vectors that aren't functions. Any help would be appreciated! Thanks! ~Matt

Below is a HW problem which I believe is correct. Can you guys take a look and advise? Which of the following are linear transformations A) L(x,y,z)= (0,0) B) L(x,y,z)= = (1,2,-1) C) L(x,y,z)= ( x^2 +y , y-z) To prove these relationships are linear transformations, they must satisfy...

L: R^2=>R^2 is defined by L(x,y)= (x+2y), (2x-y) let S be the natural basis for R^2 and T=(-1,2), (2,0). T is another basis for R^2. Find the matrix representing L with respect to A) S B) S and T C) T and S D) T E) Compute L L(1,2) using the definition...

1. Find out which of the transformations are linear. For those that are linear, determine whether they are isomorphisms. T(f(t)) = f'(t) + t^2 from P2 to P2 2. To be linear, T(f(t)+g(t))=T(f(t)) + T(g(t)), kT(f(t))=T(f(kt)) 3. After testing for linearity, I am thinking that the...

Homework Statement Let B be a basis of R^2 consisting of the vectors <5,2> and <1,5> and let R be the basis consisting of <2,3> and <1,2> Find a matrix P such that [x]_r=P[x]_b for all x in R^2 Homework Equations Ax=B? The Attempt at a SolutionI attempted by using Ax=B as a format to solve...

Homework Statement Derive a formula for T. T([1 1]^T)=[2 -1]^T and T([1 -1]^T)=[0 3]^T (...^T=transpose and T(...)=Linear Transformation Homework Equations T(c1v1+...+cnvn)=c1T(v1)+...+cnT(vn) The Attempt at a Solution The solutions manual's method and the method I am...

Let P,Q be self-adjoint linear transformations from V to V, Q is also positive-definite. Deduce that there exist scalars λ1 , . . . , λn and linearly independent vectors e1 , . . . , en in V such that, for i, j = 1, 2, . . . , n: (i) P ei = λi Qei ; (ii) <P ei , ej > = δi j λi ; (iii) <Qei ...

Homework Statement T: V-> V, dimV = n, satisfies the condition that T2 = T 1. Show that if v \in V \ {0} then v \in kerT or Tv is an eigenvector for eigenvalue 1. 2. Show that T is diagonalisable. Homework Equations The Attempt at a Solution I have shown in an earlier part...

Homework Statement The matrix A =(1,2,3;4 5 6) defines a linear transformation T: R^3-->R^2 . Find the transformation matrix for T with respect to the basis (1,0,1),(0,2,0),(-1,0,1) for R^3 and the basis (0,1),(1,0) for R^2. Homework Equations - The Attempt at a Solution I have no...

Homework Statement Determine if the transformation T: R^{2}\rightarrow R^{2} is linear if T(x, y)= (x+1, 2y) Homework Equations 1. T(u + v) = T(u) + T(v) 2. T(c*u) = cT(u) 3. T(0) = 0 The Attempt at a Solution I believe I have to use the above provided equations to determine...

Homework Statement Let u = (1,2), v = (3,1) and T: R^{2}\rightarrow R be a linear transformation such that T(u)= 4 and T(v)= 5. What is T(-3, 4)? (Hint: Write (-3,4) as a linear combination of u and v.) Homework Equations T(x)= b The Attempt at a Solution I don't really know where...

Given linear transformations S: Rn --> Rm and T: Rn --> Rm, show the following: a) S+T is a linear transformation b) cS is a linear transformation I know that since both S and T are linear transformations on their own, they satisfy the properties for being a linear transformation, which is that...

Homework Statement Let V be a subset of R2 and some fixed 1-dimensional subspace of R2. F:R2->R2 by F(v) = v if v is in V, 0 otherwise Prove that F is not a linear transformation. Homework Equations The Attempt at a Solution Just wondering if i got it right, i don't want to...

Homework Statement (124) If a linear transformation T : R3 -> R5 is one-to-one, then (a) Its rank is five and its nullity is two. (b) Its rank and nullity can be any pair of non-negative numbers that add up to five. (c) Its rank is three and its nullity is two. (d) Its rank is two and...

Homework Statement Two questions; 1. Let v1 = [-3, -4] and v2 = [-2, -3] Let T: R^2 -> R^2 be the linear transformation satisfying T(v1) = [29, -35] and T(v2) = [22, -26] Find the image of the arbitrary vector [x, y] T[x,y] = [ _ , _ ] 2. The cross product of two vectors in...

Let phi(u,v)=(u-2v,-v) is this a R^2->R^2 a linear transformation? I know that there must be two rules that must be met in order to be a linear transformation, after doing the first part, it seems that it may be linear. But I do not know how to show whether or not the second rule is...

Homework Statement Let T:U \rightarrow V be a linear transformation, and let U be finite-dimensional. Prove that if dim(U) > dim(V), then Range(T) = V is not possible. Homework Equations dim(U) = rank(T) + nullity(T) The Attempt at a Solution I almost think there must be a typo in the book...

Homework Statement I need to prove this formula, but I'm not sure how to prove it.[T]C = P(C<-B).[T]B.P(C<-B)-1 whereby B and C are bases in finite dimensional vector space V, and T is a linear transformation. Your help is greatly appreciated! Homework Equations T(x)=Ax [x]C=P(C<-B)[x]B...

Hi I am trying to do a math assignment and I am finding it really difficult. Assume you have a linear transformation from T: P3(R) --> R4 What relevance is there to applying the transformation to the basis elements of P3(R), ie: T(1), T(x), T(x^2), T(x^3)? Why is this subset special...

Homework Statement Let T:P[SUB]2 -> P[SUB]2 be the linear operator by T(a[SUB]0 +a1x + a[SUB]2x = a[SUB]o + a[SUB]1 (x - 1) + a[SUB]2 (x-1)[SUP]2 Homework Equations part a ask to find the matrix [T]B - did, see below part b ask to verify matrix [T]B satisfies every vector for [T]B [X]B...

Homework Statement Let V be a finite dimensional vector space over the field F and let S and T be linear operators on V. We ask: When do there exist ordered bases B and B' for V such that [S]B = [T]B'? Prove that such bases exist only if there is an invertible linear operator U on V such that T...

I am studying invariance, and I came across this dilemma. Suppose we have a subspace with the basis <v1, v2> of the subspace (lets say U2) and we were to map v=c1v1+c2v2 and we let c2=0. Now c1T(v1)+c2T(v2)=k1c1v1+0*T(v2)= k1c1v1. I am doing a proof and need to know what the question means by...

Homework Statement Let P3 be the space of all polynomials (with real coefficients) of degree at most 3. Let D : P3 -> P3 be the linear transformation given by taking the derivative of a polynomial. That is D(a + bx + cx2 + dx3) = b + 2cx + 3dx2: Let B be the standard basis {1; x; x2; x3}...

I thought this problem was pretty straightforward, but I can't seem to match the answers in the back of the book. The problem is: Find the determinant of the following linear transformation. T(v) = <1, 2, 3> x v (where the x means cross product) from the plane V given by x + 2y +...

Homework Statement Let B={b1,b2} be a basis for R2 and let T be the linear transformation R2 to R2 such that T(b1)=2b1+b2 and T(b2)=b2. Find the matrix of T relative to the basis B. The Attempt at a Solution I know that the matrix I'm looking for needs to be 2x2 and that the standard matrix...

I've recently come to the conclusion that i need to learn matrices. I read that matrices correspond to linear transformations and that every linear transformation can be represented by matrices, but what are linear transformation, and how do you represent it by a matrix?

Notations: L(V,W) stands for a linear transformation vector space form vector space V to W. rk(?) stands for the rank of "?". Question: Let τ,σ ∈L(V,W) , show that rk(τ + σ) ≤ rk(τ) + rk(σ). I want to know wether the way I'm thinking is right or not, or there's a better explanation...

I'm struggling to grasp what should be a trivial property of singular value decomposition. Say that I have a linear transformation T that is non-singular (i.e. T^{-1} exists) and relates matrices A and B: B = T A or A = T^{-1} B What I would like to know is how the singular values...

The problem is T(x + yi) = x - yi Show that this is a linear transformation and find the matrix of the transformation using the following basis (1+i, 1-i) ARGH I am having trouble with the complex numbers for some reason! To show that it is linear I have to show T(x + yi...

Homework Statement T: P2 --> P2 be a linear transformation defined by T(p(x)) = xp'(x) where ' is the derivative Describe the kernal and range of T and are any of the following polynomials in the range and or in the kernal of T? 2 x2 1 - x Homework Equations power rule (for...

I'm having some difficulty understanding how to perform linear transformations on matrices. I understand the definition but not how to perform the operations. I'm going to give a few examples from my book: Suppose that T: R^2 \longrightarrow R^2 is a linear transformation such that...

1. Suppose V,W are vector spaces over a field F and that T: V ---> W is a linear transformation. Show that for any v belonging to V that T(-v) = -T(v) 2. -T(v) denotes the additive inverse of T(v) 3. I think I'm really overcomplicating it =/ But i have 0v = T( v - v ) = T(v) +...

A = \left[\begin{array}{ccccc} 1 & -1 \\ 2 & 5 \\ 3 & 4 \end{array}\right] Let T_{A}: R^2 \rightarrow R^3 be the matrix transformation that maps a 2 \times 1 column vector x in R2 into the 3 \times 1 column vector Ax in R3. The relationship can be expressed as TA(x) = Ax Find a vector...