Consider the linear transformation T which transforms vectors x C) in the y-axis. a) Express the...

Consider the linear transformation T which transforms vectors x C) in the y-axis. a) Express the vector X = T(x), the result of the linear transformation T on x in terms of the components x and y of X. by reflection [10 Marks] b) Find a matrix T such that T(x) = TX, using matrix multiplication. Calculate the matrix product T2 represent? Explain geometrically (or logically) why it should be this. c) T T . What linear transformation does this a b d) Let A be a 2 x 2 matrix. Calculate the three matrix products TA, AT and TAT. For each, give a simple short description, in words concerning the rows and columns of A (say), of the result of the calculation to produce a new matrix from A.

Expert Solution

Colby Messinger answered 2 years ago

Consider the linear transformation T: R^4 to R^3 defined by T(x, y, z, w) = (x...

Consider the linear transformation T: R^4 to R^3 defined by T(x, y, z, w) = (x +2y +z, 2x +2y +3z +w, x +4y +2w) a) Find the dimension and basis for Im T (the image of T) b) Find the dimension and basis for Ker ( the Kernel of T) c) Does the vector v= (2,3,5) belong to Im T? Justify the answer. d) Does the vector v= (12,-3,-6,0) belong to Ker? Justify the answer.

A linear transformation from R3-R4 with the V set of vectors x, where T(x)=0, is V...

A linear transformation from R3-R4 with the V set of vectors x, where T(x)=0, is V a subspace of R3?

Consider the linear transformation T: R2x2 -> R2x2 defined by T(A) = AT - A. Determine...

Consider the linear transformation T: R2x2 -> R2x2 defined by T(A) = AT - A. Determine the eigenvalues of this linear transformation and their algebraic and geometric multiplicities.

Consider the linear transformation T : P2 ? P2 given by T(p(x)) = p(0) + p(1)...

Consider the linear transformation T : P2 ? P2 given by T(p(x)) = p(0) + p(1) + p 0 (x) + 3x 2p 00(x). Let B be the basis {1, x, x2} for P2. (a) Find the matrix A for T with respect to the basis B. (b) Find the eigenvalues of A, and a basis for R 3 consisting of eigenvectors of A. (c) Find a basis for P2 consisting of eigenvectors for T.

(a) Find a linear transformation T : R2→R2 that (i) maps the x1-axis to itself, (ii)...

(a) Find a linear transformation T : R2→R2 that (i) maps the x1-axis to itself, (ii) maps the x2-axis to itself, and (iii) maps no other line through the origin to itself. For example, the negating function (n: R2→R2 defined by n(x) =−x) satisfies (i) and (ii), but not (iii). (b) The function that maps (x1, x2) to the perimeter of a rectangle with side lengths x1 and x2 is not a linear function. Why? For part (b) I can't...

2) Consider the Lorentz transformation that maps (x,t) into (x',t'), where t and x are both...

2) Consider the Lorentz transformation that maps (x,t) into (x',t'), where t and x are both in time units (so x is really x/c) and all speeds are in c units also. a) Show that the inverse of the Lorentz transformation at v is the same as the Lorentz transformation at -v. Why is that required? b) Show that x^2 - t^2 = x'^2 - t'^2, i.e., that x^2 - t^2 is invariant under Lorentz transformation

Consider the linear transformation T : P1 → R^3 given by T(ax + b) = [a+b...

Consider the linear transformation T : P1 → R^3 given by T(ax + b) = [a+b a−b 2a] a) find the null space of T and a basis for it (b) Is T one-to-one? Explain (c) Determine if w = [−1 4 −6] is in the range of T (d) Find a basis for the range of T and its dimension (e) Is T onto? Explain

Consider a system with the input/output relationship y(t) = x(t)cos(15πt). (a) Is this system (i) linear,...

Consider a system with the input/output relationship y(t) = x(t)cos(15πt). (a) Is this system (i) linear, (ii) causal, (iii) stable, (iv) memoryless, (v) time-invariant, and (vi) invertible. Justify each answer with a clear mathematical argument. (b) Find the Fourier Transform Y (f) of y(t) in terms of the transform X(f) of x(t). Repeat problem (2) for the system with the input-output relationship y(t) =R1 τ=0(1−τ)2x(t−τ)dτ.

Consider the functionT:R3→R3defined byT(x,y,z) = (3x,x−y,2x+y+z).(i) Prove thatTis a linear transformation. (T’nin bir lineer d ̈on...

Consider the functionT:R3→R3defined byT(x,y,z) = (3x,x−y,2x+y+z).(i) Prove thatTis a linear transformation. (T’nin bir lineer d ̈on ̈u ̧s ̈um oldu ̆gunu g ̈osteriniz)(ii) Find the representing matrix ofTrelative to the basisβ={α1= (1,0,0),α2= (1,1,0),α3=(1,1,1)}ofR3.(R3 ̈unβ={α1= (1,0,0),α2= (1,1,0),α3= (1,1,1)}tabanına g ̈oreTnin matris g ̈osterimini bulunuz.)

1. Let X and Y be non-linear spaces and T : X -->Y. Prove that if ...

1. Let X and Y be non-linear spaces and T : X -->Y. Prove that if T is One-to-one then T-1 exist on R(T) and T-1 : R(T) à X is also a linear map. 2. Let X, Y and Z be linear spaces over the scalar field F, and let T1 ϵ B (X, Y) and T2 ϵ B (Y, Z). let T1T2(x) = T2(T1x) ∀ x ϵ X. (i) Prove that T1T2 ϵ B (X,Y) is also a...

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