Linear transformation Definition and 437 Threads

Homework Statement So there's a linear transformation T: ℝ3 → ℝ4, standard matrix A that satisfies det(A e1) = 5, det (A e2) = 4, det (A e3) = 5 and det (A e4) = 5 If S is the unit sphere, find the 3-dimensional volume of T(S). Homework Equations Volume of sphere = 4/3 * pi * r^3...

Hi We have a linear transformation g : ℝ^2x2 → ℝ g has U as kernel, U: the 2x2 symmetric matrices (ab) (bc) A basis for U is (10)(01)(00) (01)(10)(01)I thought this would be easy but I've been sitting with the problem for a while and I have no clue on how to solve it...

Homework Statement Define L: R(mxm) to R(nxn). If L(A)=L(B), prove or disprove that det(A)=det(B). Homework Equations The Attempt at a Solution I think I can prove that this is true. L(A)=L(B) means that L(A)-L(B)=L(A-B)=0. Now let C be the matrix representation of L. We...

Let T:V->V be a linear operator on an n-dimensional vector space. Prove that exactly one of the following statements holds: (i) the equation T(x)=b has a solution for all vectors b in V. (ii) Nullity of T>0

The problem is attached. The problem is "find a basis for the range of the linear transformation T." p(x) are polynomials of at most degree 3. R(T)={p''+p'+p(0) of atmost degree 2} This is pretty much as far as I got. I'm not sure how to do the rest. I'm thinking of picking a...

I attached the problem, idk if it's really easy or If I'm doing it all wrong. Since T is a linear transformation T(u+v)=T(u)+T(v)=w+0=w?

Suppose that T1: V → V and T2: V → V are linear operators and {v1, . . . , vn} is a basis for V . If T1(vi) = T2(vi ), for each i = 1, 2, . . . , n, show that T1(v) = T2(v) for all v in V . I don't understand this question. They said If T1(vi) = T2(vi ), for each i = 1, 2, . . . , n...

I attached the problem. I'm not sure if I'm misinterpreting the question, but this problem seems really easy, which is usually not the case with my class. for part a) isn't that just the coefficient matrix of the right hand side? This makes A: 1 -2 3 1 0 2 for part b) T(e1)=T[1...

The problem statement has been attached. To show that T : V →R is a linear function It must satisfy 2 conditions: 1) T(cv) = cT(v) where c is a constant and 2) T(u+v) = T(u)+T(v) For condition 1) T(cv)=∫cvdx from 0 to 1 (I don't know how to put limits into the integral...

Let A be a nxn matrix corresponding to a linear transformation. Is it true that A is invertible iff A is onto? (ie, the image of A is the entire codomain of the transformation) In other words, is it sufficient to show that A is onto so as to show that A is invertible? That was what my...

Why are we interested in looking at the kernel and range (image) of a linear transformation on a linear algebra course?

Homework Statement Let A \in M_n(F) and v \in F^n. Let v, Av, A^2v, ... , A^{k-1}v be a basis, B, of V. Let T:V \rightarrow V be induced by multiplication by A:T(w) = Aw for w in V. Find [T]_B, the matrix of T with respect to B. Thanks in advance Homework Equations...

How important are linear transformations in linear algebra? In some texts linear transformations are introduced first and then the idea of a matrix. In other books linear transformations are relegated to being an application of matrices. What is the best way of introducing linear transformation...

How important are linear transformations in linear algebra? In some texts linear transformations are introduced first and then the idea of a matrix. In other books linear transformations are relegated to being an application of matrices. What is the best way of introducing linear transformation...

Can we think of a linear transformation from R^m-->R^n as mapping scalars to vectors? Let me say what I mean. Say we have some linear transformation L from R^m to R^n which can be represented by a matrix as follows: L=[ a11x1+a12x2+...+a1mx m a21x1+... . . . anmx1+...+ anmxm...

Homework Statement Give an example of a linear vector space V and a linear transformation L: V-> V that is 1.injective, but not surjective (or 2. vice versa) Homework Equations -If L:V-> V is a linear transformation of a finitedimensional vector space, then L is surjective, L is...

Homework Statement The vector A has length 8.5, and makes an angle of 5pi/19 with the x-axis. The vector B has length 6, and makes an angle of 8pi/19 with the x-axis. Find the matrix which rotates and dilates vector into vector . Homework Equations Rotation matrix in...

Hi, I'm not sure how else to phrase this.Let's say I have a linear transformation from R3 to R2. Let's assume in both spaces, I am using the standard topology with the standard euclidean distance metric. Does this mean that open sets in R3 will be mapped to open sets in R2 under this...

Linear transformation T:\,\mathbb{R}^3\,\to\,\mathbb{R}^4 Find the standard matrix A for T T\left(x_1,x_2,x_3\right)\,=\,\left(x_1 + x_2 + x_3, x_2 + x_3, 3x_1 + x_2, 2x_2 + x_3\right) \mathbf{v}\,=\,\begin{bmatrix} x_1\\ x_2\\ x_3...

Homework Statement Let V = Span{(1,1,0), (1,2,3)}. Define a linear transformation L: V => R^3 by L(1,1,0) = (1,0,0) and L(1,2,3) = (0,1,0). For any (x,y,z) element of V find L(x,y,z) Homework Equations The Attempt at a Solution It seems like there should be some straightforward...

Homework Statement Greetings, I have been stuck with this problem for a while, I thought maybe someone could give me some advice about it. Thanks a lot in advance. If T is a linear transformation that goes from R^2 to R^2 given that T(v1)= -2v2 -v1 and T(v2)=3v2. and B =...

Homework Statement Given a linear transformation f:V -> V on a finite-dimensional vector space V, show that there is a positive integer m such that im(f^m) and ker(f^m) intersect trivially. Homework Equations The Attempt at a Solution Observe that the image and kernel of a linear...

Homework Statement Find a kernel and image basis of the linear transformation having: \displaystyle T:{{\mathbb{R}}^{3}}\to {{\mathbb{R}}^{3}} so that \displaystyle _{B}{{\left( T \right)}_{B}}=\left( \begin{matrix} 1 & 2 & 1 \\ 2 & 4 & 2 \\ 0 & 0 & 0 \\ \end{matrix} \right)...

Hello, I'm given this linear transformation and I'm asked to do the typical calculations (kernel, image, dimensions, etc.) but there's one thing I'm not sure I understand, here's the exercise: f(1,0,0)=(-1,-2,-3) f(0,1,0)=(2,2,2) f(0,0,1)=(0,1,2) a) Is f invertible? b)Find a basis of...

Sorry I feel like an idiot for asking this but why is part c and b not a linear transformation? The origin would still be (0,0) and it's an expression in x and y terms so I'm confused? thanks

Bases of a Linear transformation (Kernel, Image and Union ? http://dl.dropbox.com/u/33103477/1linear%20tran.png For the kernel/null space \begin{bmatrix} 3 & 1 & 2 & -1\\ 2 & 4 & 1 & -1 \end{bmatrix} = [0]_v Row reducing I get \begin{bmatrix} 1 & 0 & \frac{7}{9} & \frac{-2}{9}\\ 0 & 1...

Homework Statement Let T be the linear transformation from P2 (R) to P3 (R) defined by T(f)=14\int_{0}^{x}f(t)dt + 7x.f'(x) for each f(x)=ax^{2}+bx+c Determine a basis {g1, g2, g3} for Im(T). Homework Equations as above The Attempt at a Solution I evaluated the...

Homework Statement Let A be an 3x3 matrix so that A^3 = {3x3 zero matrix}. Assume there is a vector v with [A^2][v] ≠ {zero vector}. (a) Prove that B = {v; Av; [A^2]v} is a basis. (b) Let T be the linear transformation represented by A in the stan- dard basis. What is [T]B? Homework...

http://dl.dropbox.com/u/33103477/linear%20transformations.png My solution(Ignore part (a), this part (b) only) http://dl.dropbox.com/u/33103477/1.jpg http://dl.dropbox.com/u/33103477/2.jpg So I have worked out the basis and for the kernel of L1 and image of L2, so I have U1 and U2...

I am working on a problem dealing with transformations of a vector and finding the basis of its kernel. Now I have worked out everything below but after reading the definitions I am a bit confused, hence just want verification if the procedure I am following is correct. My transformed matrix is...

Homework Statement Let A_{2x2} have all entries=1 and let T: M_{2x2}\rightarrowM_{2x2} be the linear transformation defined by T(B)=AB for all B\inM_{2x2} Find the matrix C=[T]s,s, where S is the standard basis for M_{2x2} My solution: Standard basis for M_{2x2}={(1,0),(0,1)}...

Homework Statement Is T(X,Y)->(X,Y,1) a linear transformation? where X and Y are defined R2 column vectors. Homework Equations Attempt to prove T(cX+Y)=cT(X)+T(Y) Consider T(cx1+y1,cx2+y2)->(cx1+y1,cx2+y2,1) The Attempt at a Solution RS=cT(x1,y1)+T(x2,y2)->c(x1,y1,1)+(x2,y2,1)...

Homework Statement Let T: R3-->R3, defined by T(x)= a x x Give the standard matrix A of T, and explain why A is skew-symmetric. Homework Equations They define u x v as u x v=(det [u2 u3/ v2 v3], det [u3 u1 /v3 v1], det [u1 u2/ v1 v2]) For any vectors u,v,w in R3...

Having a bit of trouble with this one. Can anyone help? Many thanks. Homework Statement Q. L is the line x - y + 1 = 0. f is the transformation f: (x, y) ---> (x', y') where: x' = 2x - y & y' = y. Find f(L) and investigate if f(L) is parallel to L. Homework Equations The...

i am having trouble with some homework problems in my linear algebra course... the book is brescher and the teacher is sort of a rambling nutcase whose presentation of material is anything but 'linear', and very difficult for me to follow. similarly the book contains problems that i can't seem...

Homework Statement Assume that T:C^0[-1,1] ---> ℝ. assume that T is a linear transformation that maps from the set of all continuous functions to the set of real numbers. T(f(x)) = ∫f(x)dx from -1 to 1. is T one to one, is it onto, is it both or is it neither. Homework Equations definition...

Homework Statement Let s be the linear transformation s: P2→ R^3 ( P2 is polynomial of degree 2 or less) a+bx→(a,b,a+b) find the matrix of s and the matrix of tos with respect to the standard basis for the domain P2 and the standard basis for the codomain R^3 The Attempt at a...

I have got t: P3 → P3 p(x) → p(x) + p(2) and s: P3 → P2 p(x) → p’(x) thus s o t gives P3→ P2gives p(x) → p’(x) next part says : use the composite rule to find a matrix representation of the linear transformation s o t when t: P3 → P3 p(x) : p(x) + p(2) and...

Homework Statement The Attempt at a Solution I: P(R) → P(R) such that I(a0+a1x + ... + anxn) = 0 + a0x + (a1x2)/2 + ... + (an xn+1)/(n+1) Clearly this is just integration such that c = 0. It is easily shown that integration is a linear transformation, so I conclude that I is a...

Homework Statement Suppose A is an mxn matrix and b is a vector in R^m. Define a function T:R^n --> R^m by T(x) = Ax + b. Prove that if T is a linear transformation then b=0. Homework Equations For the second part of the question, a transformation is linear if: 1) T(u+v) = T(u) +...

Not too big of a question, it's more so just handling a certain presentation. T: M2x2-->M2x2 defined by: T(A) = (A + AT)/2. So my question is, how do I handle the fraction considering it will be a matrix?

Why would you want to use a matrix for a linear transformation? Why not just use the given transformation instead of writing it as a matrix?

1. Say you have a linear transform from A to B, and where A has a higher dimension than B. How do you show that the kernel of the transform has more than one element (i.e. 0)? Also, if B has a higher dimension than A, then how to show that the transform isn't surjective? 2. The attempt at a...

Is there a short and simple proof of the Nullity - Rank Theorem which claims that if T: U->V is a linear transformation then rank(T)+Nullity(T)=n where n is the n dimension vector space U.

Hi, My question is to show that the linear transformation T: M2x2(F) -> P2(F) defined by T (a b c d) = (a-d) | (b-d)x | (c-d)x2 is surjective but not injective. thanks in advance.

I need help with question from homework in linear algebra. This question (linear transformation): http://i43.tinypic.com/15reiic.gif According to theorem dimensions: dim(V) = dim(Ker(T)) + dim(Im(T)). dim(Ker(T))=2. dim(V) in R^4, meaning =4. We can therefore conclude that dim(Im(T))=2. But...

Linear transformation T: R3 --> R2 Homework Statement Find the linear transformation T: R3 --> R2 such that: T(1,0,0) = (2,1) T(0,1,1) = (3,2) T(1,1,0) = (1,4) The Attempt at a Solution I've been doing some exercises about linear transformations (rotations and reflections...

I've seen in multiple sources that a linear transformation constitutes a tensor with one contravariant and one covariant index. Could someone explain to me why this is the case? I'm asking not because I have a solid understanding of tensors and am confused about this particular example; rather...

Homework Statement Let V be polynomials, with real coefficients, of degree at most 2. Suppose that T:V→V is differentiation. Find the B-matrix [T]B if B is the basis of V B = {1+x, x+x2, x} Homework Equations For T:V→V the domain and range are the same [T]B is the matrix whose i-th...

Homework Statement Let h: \mathbb{P_2} \rightarrow \mathbb{P_2} represent the transformation h(p(x)) = xp'(x) + p(1-x) for every polynomial p(x) \in \mathbb{P_2}. Find the matrix of h with respect to the standard basis \{1, x, x^2\} Homework Equations Matrix A of transformation: {\bf A}...