Verify that GLz(R) is a qroup under the matrix mu lti plication. a 3 all 2x2 invertible matrices O Show by counterexample that GLz(R) is not abelian, T choose A,B 6.t A*B#B*Aas e

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3) Let T: R» R' be a linear transformation and X = x) T(x)is defined as [x3 follows: T(x1,x2, x3) = (4x| – x2 + 2x3, 3x¡ + 2x½ - 5x3, 2x + x2 – x3) a) Find the standard matrix of T. b) Show that T is invertible by row reducing the appropriate matrix, and find the inverse of the standard matrix. c) Find a formula for T. d) Suppose a y e R' and a bɛ R' such that T'(y)= b. Give the equivalent statement relating b,T, and y.

How did they get these answeres for e and f. I have listed answers with questions but dont know how they got them

note the inverted wedge is (lambda).

We define a linear bijection, h, between R4 and H(2), the set of complex 2 x 2 hermitian matrices, by y – iz t - x t +x (t, z, y, 2) → (, y + iz We denote the matrix on the right-hand side by H. (i) Show that the matrix can be written as a linear combination of the Pauli spin matrices 01, 02, 03 and the identity matrix I2. (ii) Find the inverse map. (iii) Calculate the determinant of 2 x 2 hermitian matrix H. Discuss.

The linear tranformation L defined by L(p(x)) = -13p' – 14p" maps P, into P3. (a) Find the matrix representation of L with respect to the ordered bases = {x³, x², x, 1} and F = {x² + x +1, x + 1, 1} S = (b) Use Part (a) to find the coordinate vectors of L(p(x)) and L(g(x)) where p(x) -4x° + 7x and g(x) = x2 – 11. [L(p(x))]F [L(g(x))]F

Let operator be T: R³ → R³ with correspondence rule T(х, у, 2) %3 (у, х +z,0) a) Determine the characteristic values of T. b) Obtain the characteristic spaces of T. c) Indicate whether A is diagonalizable and, if so, obtain the matrix P that diagonalizes A.

determine wether the given set of invertible n x n matrices with real number entries is a subgroup of GL(n,R). The n x n matrices with determinant 2^k for some integer k