Linear transformation Definition and 437 Threads

Matrix of linear transformation (urgent) Identify the matrix of the transformation for the following: a) (x,y,z) = (2x-y+4z,x+y-z,x-z) b) (x,y) = (x,2x) c) (x,y,z) = (x-2y,3x-6y) Here are my attempts a) 2,-1, 4 1, 1,-1 1, 0,01 b) 1,0 2,0 c) 1,-2, 0 3,-6, 0...

First of all I would like to wish a happy new year to all of you, who have helped us understand college math and physics. I really appreciate it. Homework Statement Determine the dimension of the image of a linear transformations f^{\circ n}, where n\in\mathbb{N} and...

Homework Statement U = [Polynomial of degree 3 such that 3p(1) = p(0)] Find the basis of U and find a linear transformation T: P3 ---> R such that U is the kernel of T.Homework Equations The Attempt at a Solution The basis part is easy. 3p(1) = p(0) 3a + 3b + 3c +d = d c= -b-a Basis ...

I have two linear differential operators L_1 = D + 1 and L_2 = D - 2x^2 for L_1(L_2) = (D + 1)(D - 2x^2) = (D)(D - 2x^2) + (1)(D - 2x^2) = D(D) - D(2x^2) + D - 2x^2 = D^2 + D(1 - 2x^2) - 2x^2 does that look right? I might be making an error somewhere but my book says: L_1(L_2) = D^2...

1. Find the image of the linear transformation whose matrix is given by: 1 2 5 2 4 -3 1 0 10 -13 -7 -4 Homework Equations 3. Tried numerous times but struggle to get anywhere

Homework Statement (note; all column vectors will be represented as transposed row vectors, and matrices will be look like that on a Ti-83 or similar) L: R^3 -> R^2 is given by, L([x1, x2, x3]) = [2x1 + x2 - x3 x1 + 3x2 +2x3]* *Matrix Relevant...

Homework Statement Prove or dispove T:R^n \rightarrow R^n is a linear transformation if for every u \in R^n and for every v \in kerT , T(u) \cdot v=0 then KerT = (ImT)^{\perp}The Attempt at a Solution True. since it was given T(u) \cdot v=0 and we are dealing with all of R^n then...

Hi guys, I have two practice problems with no solutions that i was not able to figure out. If anyone could help I'd appreciate it. Question 1 Homework Statement Find the basis of {(x,y,z) | x + y + 2z = 0} Homework Equations None? The Attempt at a Solution I can find the...

Homework Statement Let V be a finite-dimensional real inner product space with inner product < , >. Let L:V->R be a linear map. Show that there exists a vector u in V such that L(x) = <x,u> for all x in V. 2. The attempt at a solution It seems really simple but I just can't phrase...

What does the transpose of: example, [1 0 -1]? how can you transpose that? For example the L([a b c]*) --> [a + b a - c]* how do i show that this is a linear transformation? *this is transposed.

Homework Statement Let T:Rn to Rm be a linear transformation that maps two linearly independents vectors {u,v} into a linearly dependent set {t(u),T(v)}. Show that the equation T(x)=0 has a nontrivial solution. Homework Equations c1u1 + c2v2 = 0 where c1,c2 = 0 T(c1u1 + c2v2) = T(0)...

Homework Statement Given a linear transformation F: R^3 --> R, F(x,y,z) = 3x-2y+z, find I am (F) and dim (Im (F)) Homework Equations I have found that dim(ker F) = 2 and from the theorem dim (V) = dim (Ker F) + dim (Im F), I know dim (V) = 3, so dim (Im F) = 1. The Attempt at...

Homework Statement Let T:\Re^{n}\rightarrow\Re^{m} and let S={u,v,w}\in\Re^{n}. If S is linearly dependent, show that {T(u), T(v), T(w)} is also linearly dependent. Homework Equations N/A The Attempt at a Solution Since S\in\Re^{n} then S`\in\Re^{m}. Not sure where to go from here

Homework Statement Hello! Prove: A(\vec{a}+\vec{b}) = A\vec{a} + A\vec{b} Where A is a matrix and T (in the following section) is a transformation. Homework Equations T(\vec{a}) + T(\vec{b}) = T(\vec{a}+\vec{b}) T(\vec{a}) = A\vec{a} T(\vec{b}) = A\vec{b} The Attempt at a...

Homework Statement Let X be the vector space of polynomial of order less than or equal to M a) Show that the set B={1,x,...,x^M} is a basis vector b) Consider the mapping T from X to X defined as: f(x)= Tg(x) = d/dx g(x) i) Show T is linear ii) derive a matrix...

Homework Statement Mean of 25 and standard deviation of 5. Rescale the test using linear transformation so that the mean is 100? and the standard deviation is 20... Homework Equations xnew=a+bx The Attempt at a Solution I don't know...15+4x? I really don't understand how to...

Homework Statement There is a linear transformation T from P1 to P1 where P1 is the set of all polynomials of degree at least 1. T(1 + 2x) = 2 + 4x and T(4 + 7x) = -2 + 2x Find T(-3 - 5x).Homework Equations T(1 + 2x) = 2 + 4x T(4 + 7x) = -2 + 2x The Attempt at a Solution Basis B1 = [1, 2]...

Hey all, I'm trying to find an orthogonal complement (under the standard inner product) to a space, and I think I've found the result mathematically. Unfortunately, when I apply the result to a toy example it seems to fail. Assume that A \in M_{m\times n}(\mathbb R^n), y \in \mathbb R^n and...

Homework Statement How do I find the bases for both the kernel and range of this linear transformation? Let T: R4 ----> R4 be the linear transformation that takes [1101] and [1011] to [2304] and takes [1110] and [0111] to [3120] a. Find the bases for both the kernel and the range of...

I'm hoping I can get some help with the following question: Does definite integration (from x = 0 to x = 1) of functions in Pn correspond to some linear transformation from Rn+1 to R? OK, well my original answer was yes, but the textbook says "no, except for P0" which I do not understand...

Homework Statement Let T: M22→M22 be a LT defined by T(A)=AB where B=[3,2 2,1] Determine if T is invertible with respect to standard bases B=C={e11,e12,e21,e22}. If so, use (equation below) to find T^-1. Homework Equations [[T^-1 [AB]]C = [T^-1]B to C matrix [AB]B (at least...

Homework Statement Let L: R^3 -> R^3 be a linear transformation such that L(i) = [1 2 -1], L(j) = [1 0 2] and L(k) = [1 1 3]. Find L([ 2 -1 3)]. All the numbers in [ ] should be vertical, but I don't know how to set that up. Homework Equations The Attempt at a Solution...

Homework Statement T(a+bx+cx^2) = [b+c a-c] What is Ker(T) Homework Equations I don't the relevant equation(s). I know that the definition of the kernal of a LT is the set of all vectors that are mapped to 0 by T. The Attempt at a Solution What...

Hi, I am trying to find an orthogonal transformation that maps the point (0,5) to the point (3,4). Now, I found that the transformation matrix M for a reflection in the line y=mx is as follows: M = \left( \begin{array}{cc} cos(2\theta) & sin(2\theta)\\ sin(2\theta) & -cos(2\theta) \end{array}...

Hey, i have an assignment in MATLAB class which is Let L be a linear transformation such that L(1)=(2 -1)' L(1-x)=(1 0)' L(1+x^2)=(1 1)' L(1+x^3)=(1 2)' Determine a matrix in domain such that with the canonical in range, the matrix that represents L has two null columns. I don't know...

Homework Statement Verify the linear transformation & find the standard matrix A T:R2->R2, T(x,y) = (x1+5,x2) Homework Equations The Attempt at a Solution so i have to verify addition and multiplication T(u+v) = ((u1+v1)+5,(u2+v2) Does this fail.. it seems i will never be...

Homework Statement Let T be the linear transformation T: M2x2-->M2x2 given by T([a,b;c,d]) = [a,b;c,d][0,0;1,1] = [b,b;d,d] Find bases (consisting of 2x2 matrices) for the image of T and the nullspace of T. Homework Equations Standard basis of a 2x2 matrix...

L: R^3 -> R^3 is a linear transformation defined by L(v) =A(v) A is given as -1 2 0 and w= 1 1 1 1 2 2 -1 1 -1 is w in the range of L? My understanding is that if a vector exists such that the...

Homework Statement Consider a linear transformation L from Rm to Rn. Show that there is an orthonormal basis {v1,...,vm} of Rm to Rn such that the vectors {L(v1),...,L(vm)} are orthogonal. Note that some of the vectors L(vi) may be zero. HINT: Consider an orthonormal basis {v1,...,vm} for...

[b]1. Find the kernel and range of the linear transformation. Indicate whether its 1-1, onto, both or neither [b]2. U: P2-----> R^2 defined by U(f(x)) = [f(1), f ' (1)] [b]3. To me by looking at the problem, it seems as if its going to be 1-1. As for solving this problem...I AM...

Do all linear transformations are matrix transformation? In a book by David C Lay, he wrote on page 77 that not all linear tranformations are matrix transformations and on page 82 he wrote that very linear transformation from Rn to Rm is actually a matrix transformation. I know that every matrix...

One of the topics in my linear algebra course is kernel and range of a linear transformation. I have a firm understanding of what the kernel is: the set of vectors such that it maps all inputs to the zero vector. Range, however, remains nebulous to me. My textbook says that the range is "THe...

Homework Statement L(p(t)) = t*dp/dt + t^2*p(1) If p(t) = a*t^2 + b*t + c, find a basis for the kernel of L. Homework Equations None. The Attempt at a Solution I know that L(a*t^2 + b*t + c) = 0, so that would mean that the derivative needs to be zero and p(1) needs to be zero. This...

Homework Statement Linearly speaking the Kernel of T is a ? Homework Equations I solved kernel of T to equal {<-5,3,1>} The Attempt at a Solution So is Kernel of T is a plane?

Homework Statement The following vectors form an ordered basis E = [v1, v2] of the subspace V = span(v1,v2): v1 = (1,2,1)^T , v2 = (3,2,1)^T. The vector v = (24,-8,-4)^T belongs to the subspace V. Find its coordinates (c1,c2)^T = [v]E relative to the ordered basis E = [v1,v2]. Homework...

Homework Statement Find the matrix of the transformation: T: R^{2} \rightarrow R^{2x2} \[ T(a,b) = \left[ {\begin{array}{cc} a & 0 \\ 0 & b \\ \end{array} } \right] \] Homework Equations The Attempt at a Solution I choose the standard bases for R^{2} and R^{2x2}...

Homework Statement Let V be a vector space over a field F = R or C. Let W be an inner product space over F. w/ inner product <*,*>. If T: V->W is linear, prove <x,y>' = <T(x),T(y)> defines an inner product on V if and only if T is one-to-one Homework Equations What we know, W is an inner...

Homework Statement Linear transformation T: P2->P2 T(f) = -5f + 8f' Need to find detA (A is a matrix of T) Homework Equations T(f) = Af The Attempt at a Solution The basis of P2 is B={1, x, x2}. Some polynomial f with respect to B looks like this in general: (a, b, c)T right...

c . L(A) = A + I L(alpha A + beta B) = +(alpha A + I + beta B, I) and +(alpha A + I + beta B, I) = alpha A + beta B + 2*I alpha L(A) + beta L(B) = alpha (A + I) + beta (B + I) and alpha (A + I) + beta (B + I) = alpha A + beta B + I (alpha + beta) Are these steps correct for the linear...

Homework Statement Given that "If T(Ta)=0, then Ta=0", can we say that the linear transformation on V is nonsingular? Homework Equations The Attempt at a Solution Since what the statement implies is that T has only zero subspace of V as its null space, can we not say that it's...

Homework Statement A robot arm in a xyz coordinate system is doing three consecutive rotations, which are as follows: 1) Rotates (Pi/4) rad around the z axis 2) Rotates (Pi/3) rad around the y axis 3) Rotations -(Pi/6) rad around the x axis Find the standard matrix for the (combined)...

Homework Statement T: V --> W is a linear transformation where V and W are finite dimensional. If dim V is less than or equal to dim W, then T is one-to-one. True or false? Homework Equations The Attempt at a Solution First of all, I'm assuming that im(T) = W. Is that correct...

I think I've solved this problem, but the examples in my textbook are not giving me any indication as to whether my reasoning is sound. Homework Statement Is the transformation T(M) = M\left[ \begin{array}{cccc} 1 & 2 \\ 3 & 6\end{array} \right] from \mathbb{R}2x2 to \mathbb{R}2x2 linear...

Homework Statement Suppose T : V --> W is a linear transformation and one-to-one. Show, if ||.|| is a norm on W, then ||x|| =||T(x)|| is a norm on V. (V and W are vector spaces) Homework Equations T is linear, so T(x+y)= T(x) + T(y) and T(ax)= aT(x) T is one-to-one, so T(x)=T(y)...

Homework Statement F:R^2 to R^2 defined by F(x)= x1+x2 1 Where x= x1 x2 Homework Equations Must satisfy these conditions: T(u+v)=T(u)+T(v) T(au)=aT(u) The Attempt at a Solution I said u= u1 u2 v= v1 v2 u+v= u1+u2 v1+v2 then F(u+v)= (u1+v1) +...

The problem is as follows: Find a nonzero 2x2 matrix A such that Ax is parallel to the vector [1] [2] for all x in R2. So far I know A=[v1 v2] therefore Ax= [v1 v2][x1] [x2] = x1v1+x2v2 I know these two vectors are parallel, but I am a...

Homework Statement Show that if the matrix of a linear transformation "multiplication by a" is "A" then a is a root of the characteristic polynomial for A. Also, I am not sure how to obtain the monic polynomial of degree 3 satisfied by 2^(1/3) and by 1+2^(1/3)+4^(1/3). The Attempt...

Hi, I had to prove that if T2=T then the direct sum of im(T) and ker(T) is a vector space, and I did that, but now I am suppose to find such a T that isn't the zero operator (I'm not even sure what that is, just a transformation that makes any vector zero?) or the identity operator. Problem is I...

Homework Statement Let V be the space of polynomials with degree \leq n (dimV=n+1) i. Let D:V->V be differentiation, i.e. D: f(x) -> f'(x) What are the eigenvalues of D? Is D diagonalisable? ii. Let T be the endomorphism T:f(x) -> (1-x)2 f''(x). What are the eigenvalues of T? Is...

Homework Statement Let S(U)=V and T(V)=W be linear maps where U,V, W are vector spaces over the same field K. Prove : Homework Equations a) Rank (TS) <= Rank (T) b) Rank (TS) <= Rank (S) c) if U=V and S is nonsingular then Rank (TS) = Rank (T) d) if V=W and T is nonsingular then Rank...