Question

Let **R*= R\** {0} be the set of nonzero real
numbers. Let

G= {2x2 matrix: row 1(a b) row 2 (0 a) | a in
**R***, b in **R}**

(a) Prove that G is a subgroup of GL(2,**R**)

(b) Prove that G is Abelian

Answer #1

Let G be the set of all 2x2 matrices [a a a a] such that a is in
the reals and a does not equal 0.
Prove or disprove that G is a group under matrix
multiplication.

Let R be a relation on set RxR of ordered pairs of real numbers
such that (a,b)R(c,d) if a+d=b+c. Prove that R is an equivalence
relation and find equivalence class [(0,b)]R

1. Let A be the 2x2 matrix in M_{2x2}(C), whose first row is
(0,1) and second row is (-1,0).
(a) Show that A is normal.
(b) Find (complex) eigenvalues of A.
(c) Find an orthogonal basis for C^2, which consists of
eigenvectors of A.
(d) Find an orthonormal basis for C^2, which consists of
eigenvectors of A.

Prove or disprove: GL2(R), the set of invertible 2x2 matrices,
with operations of matrix addition and matrix multiplication is a
ring.
Prove or disprove: (Z5,+, .), the set of invertible
2x2 matrices, with operations of matrix addition and matrix
multiplication is a ring.

8. Let A = {fm,b : R → R | m not equal 0 and
fm,b(x) = mx + b, m, b ∈ R} be the group of affine
functions. Consider (set of 2 x 2 matrices) B = {[ m b 0 1 ] | m, b
∈ R, m not equal 0} as a subgroup of GL2(R) where R is
the field of real numbers..
Prove that A and B are isomorphic groups.

Let G be an Abelian group. Let k ∈ Z be nonzero. Define φ : G →
G by φ(x) = x^ k . (a) Prove that φ is a group homomorphism. (b)
Assume that G is finite and |G| is relatively prime to k. Prove
that Ker φ = {e}.

Let E⊆R (R: The set of all real numbers)
Prove that E is sequentially compact if and only if E is
compact

For which subspaces S in R^2 is there a 2x2 matrix A for which
row(A)=col(A)=S?

Prove: Let S be a bounded set of real numbers and let a > 0.
Define aS = {as : s ∈ S}. Show that inf(aS) = a*inf(S).

Find an example of a nonzero, non-Invertible 2x2 matrix A and a
linearly independent set {V,W} of two, distinct
non-zero vectors in R2 such that
{AV,AW} are distinct, nonzero and
linearly dependent. verify the matrix A in non-invertible, verify
the set {V,W} is linearly independent and verify
the set {AV,AW} is linearly
dependent

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