2. (5 points) Let T: R2 + R3 be a linear transformation with 2x1 - x2] 1-3x1 + x2 | 2x1 – 3x2 Find x = (x) <R? such that [0] -1 T(x) = (-4)
Problem 5 (25 points). Let Mat2x2(R) be the vector space of 2 x 2 matrices with real entries. Recall that (1 0.0 1.000.00 "100'00' (1 001) is the standard basis of Mat2x2(R). Define a transformation T : Mat2x2(R) + R2 by the rule la-36 c+ 3d - (1) (5 points) Show that T is linear. (2) (5 points) Compute the matrix of T with respect to the standard basis in Mat2x2 (R) and R”. Show your work. An answer with...
Define the linear transformation T:?3??4 by T(x )=Ax . Find a vector x whose image under T is b (1 pt) Let 4 5 2 -2 5 -3 2 and b-10 -7 2 1 -4 Define the linear transformation T : R3 ? R4 by T(x-Ax Find a vector x whose image under T is b. x= Is the vectorx unique? choose
Question 1.2 Let T : R3 ? R2 be a linear transformation given by T (x) = Ax, where 1 0 2 -1 1 5 1) Find a basis for the kernel of T. 2) Determine the dimension of the kernel of T 3) Find a basis for the image(range) of T. 4) Determine the dimension of the image(range) of T. 5) Determine if it is a surjection or injection or both. 2 6) Determine whether or not v |0|...
Let T: R3 → R3 be the linear transformation that projects u onto v = (9, -1, 1). (a) Find the rank and nullity of T. rank nullity (b) Find a basis for the kernel of T.
5. (7 points) Let f: R3 → R be the function f(x,y,z) = x2 + y2 +3(2-1)2 Let EC R3 be the closed half-ball E = {(x, y, z) e R$: x² + y2 +< 9 and 2 >0}. Find all the points (x, y, z) at which f attains its global maximum and minimum on E.
Let ?: ?2(R) ⟶ R3 be a linear transformation such that ?(1) = (−1, 2, −3), ?(1 + 3?) = (4, −5, 6), and ?(1 + ?2) = (−7, 8, −9). a. Show that {1,1 + 3? ,1 + ?2} is a basis for ??2(R) (7pts) b. Compute ?(−1 + 4? + 2?2). (3pts)
5. Let T: P2(R) R3 be a linear transformation such that T(1) = (-1,2, -3), T(1 + 3x) = (4,-5,6), and T(1 + x²) = (-7,8,-9). a. Show that {1,1 + 3x ,1 + x2} is a basis for P(R) (7pts) b. Compute T(-1+ 4x + 2x²). (3pts)
R is defined by T (7) = AZ mation T: R3 4. [20 marks) A linear transformation T: R with A given as follows: A= [ 1 -2 1 3 0 -21 1 6 -2 -5 J (1). (8 marks) A vector in R is given as follows = -1 determine the image of 7 under T. 12 marks) Find a vector in Rwhose image under T is the following vector 6 -17 7 = 7 L -3 or demonstrate...
LINEAR ALGEBRA (1) SECOND SEMSTER FINAL HOMEWORK (5) 31-5-2020 Question 6: a) Determine whether T : R3 → R defined by T(x,y) = xy is linear transformation or not (show all the axioms ) b) Find u.v given that | u +v1=2 and u-v||=6 c) Let V and U be vectors in R” Then prove V.U\<|||||||||| A = 2 11 - 2 3 1-27 d) suppose 4 1, V = 0,U = 2 -1 0 5 4 1) Determine whether...