5. Consider two linear transformations A and B with matrix representations 4-[4 B = 1 2...
2. Work with matrix representations of linear transformations and use knowledge of matrix properties to prove that if a EC is an eigenvalue of a linear operator T:V + V on a (finite-dimensional) inner product space V over C, then ā is an eigenvalue of the adjoint operator T* :V + V. Hint: Check that det (Tij) = det (ij) and utilize this property.
Determinants and linear transformations 4. (a) Let A be the matrix 1 -2 4 1 3 2 11 i) Calculate the determinant of A using cofactor expansion of row 3. (ii) Is A invertible? If so, give the third column of A1 (you do not have to simplify any fractions) (b) Let B be the matrix 0 0 4 0 2 8 0 4 2 1 0 0 0 7 Use row operations to find the determinant of B. Make...
136. Transformations for Different Bases. Find the matrix A that represents the linear transformation T with respect to the bases B and B'. (a) T:R3M2,2 given by T(4, 0, 2) =: -20 where B = {e1,e2, C3} and B' = {EM i = 1, 2; j = 1,2} (i.e. the standard basis for M2,2). (b) T:P3 + P3 given by T(ao + ax + a2r? + agr) = (do + a2) - (ai +203) where B, B' = {1,2,2, "}.
Consider the 2×22×2 matrix AA given by A=1−2[−5−1−1−5].A=1−2[−5−1−1−5].. Find the eigenvalues λ+λ+ and λ−λ−, larger and smaller or equal or conjugate, respectively, of the matrix AA, I am really stuck on parts b and c so any help would be greatly appreciated! (10 points) 5 Consider the 2 x 2 matrix A given by A al -}] 1 a. (2/10) Find the eigenvalues l_ and _, larger and smaller or equal or conjugate, respectively, of the matrix A, + =...
In Exercises 1-14. find the matrix representations Rg and Rr and an invertible matrix C such that R CRC for the linear transjormation T of the given vector space with the indicated ordered bases B and B' derivative of p(x); B = (x', x', x, l), B' = (1, x , x1, x' + 1) 14. T: WW, where W sp(e, xe') and T is the derivative transformation; B (e, xe*), B = (2xe", 3e* In Exercises 1-14. find the...
Consider the 2-dimensional system of linear equations -2 X' = 2 Note that the coefficient matrix for this system contains a parameter a. (a) Determine the eigenvalues of the system in terms of a (b) The qualitative behavior of the solutions depends value ao where the qualitative behavior changes. Classify the equilibrium point of the system (by type and stability) when a < ao, when a = a), and when a > ao. on the value of a. Determine a...
1) a) If A is a 4×5 matrix and B is a 5×2 matrix, then size of AB is: b) If C is a 3×4 matrix and size of DC is 2×4 matrix , then size of D is: c) True or False: If A and B are both 3 × 3 then AB = BA d) The 2 × 2 identity matrix is: I = e) Shade the region 3x + 2y > 6. f) Write the augmented matrix...
eclass.srv.ualberta.ca 2 of 2 1. Consider the matrix 3-2 1 4-1 2 3 5 7 8 (a) Find a basis B for the null space of A. Hint: you need to verify that the vectors you propose 20 actually form a basis for the null space. (Recall: (1) the null space of A consists of all x e R with Ax = 0, and (2) the matrix equation Ax = 0 is equivalent to a certain system of linear equations.)...
2. Consider the linear system -2.c + 3y + 2 = 5 4.c +9y-322 = 5 -2.c + 18y - 292 = 20 (a) Write down the augmented matrix of the linear system. (b) Find the reduced row-echelon form of your matrix from part (a). (c) Using your answer to part (b), write down the solution to the linear system. Clearly indicate which variable(s) (if any) you are using as a free variable(s).
1. Consider the matrix 12 3 4 A 2 3 4 5 3 4 5 6 As a linear transformation, A maps R' to R3. Find a basis for Null(A), the null space of A, and find a basis for Col(A), the column space of A. Describe these spaces geometrically. 2. For A in problem 1, what is Rank(A)?