Build an isomorphism between M2  (R) and P3  (R)

The factor group U(16)/(9)has an element of order 4. True of false? True O Falsc

Determine the groups of quadratic residues Qn for all n in the range 10 ≤ n ≤ 20.

Show that the function is a linear transformation. Is it also an isomorphism? Upload your proofs below. Let V be the space of infinite sequences of real numbers. T(x0, x1, x2, x3, x4..) = (x0, x2, x4, ...) from V to V (we are dropping every other term) OLinear but not an isomorphism OLinear and an isomorphism ONot Linear Choose File No file chosen

GIVEN: P₂ = { a₁ + a₁x + a₂x² ao, a₁, a₂ ≤ R} T: P₂ R, T(a₁ + a₁x + a₂x²) = a₂ For example: 7(1- x + 3x²) = 3 PROVE: T is a linear transformation. -

give the order of the element in the factor group (2,0) + <(4,4)> in (Z6 x Z8)/<(4,4)>

5) Are the following linear transformations? (Show that they are, or where they fail to be and SHOW ALL STEPS in the domain and codomain.): a) L:R3 R2: L(x1, X2, X3) = (x2, X3) b) L: P2 → P3; L(p(x)) = x² + p(x)

Consider maps π1 : G1 × G2 → G1 given by π1(g1, g2) = g1 and π2 : G1 × G2 → G2 given byπ2(g1, g2) = g2. Show that π1 and π2 are homomorphisms (You can just show one of them is a homomorphism and say the other one follows similarly.)