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For Problems 1-7, determine whether the given set
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- Consider the matrices R=[ 0110 ] H=[ 1001 ] V=[ 1001 ] D=[ 0110 ] T=[ 0110 ] in GL(2,), and let G={ I2,R,R2,R3,H,D,V,T }. Given that G is a group of order 8 with respect to multiplication, write out a multiplication table for G. Sec. 3.3,22b,32b Find the center Z(G) for each of the following groups G. b. G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of section 3.1. Find the centralizer for each element a in each of the following groups. b. G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of section 3.1 Sec. 4.1,22 22. Find an isomorphism from the octic group D4 in Example 12 of this section to the group G={ I2,R,R2,R3,H,D,V,T } in Exercise 36 of Section 3.1. Sec. 4.6,14 14. Let G={ I2,R,R2,R3,H,D,V,T } be the multiplicative group of matrices in Exercise 36 of section 3.1, let G={ 1,1 } under multiplication, and define :GG by ([ abcd ])=adbc. Assume that is an epimorphism, and find the elements of K= ker . Write out the distinct elements of G/K. Let :G/KG be the isomorphism described in the proof of Theorem 4.27, and write out the values of .arrow_forwardProblem #9: Given three linearly independent vectors V₁ = (-9,3,-9,9,-2), V2 = (-8, 10, 9, -3, 10), V3 = (-7,-9,-7, -7,0), in R5, complete this to a basis using the first two standard basis vectors which work. If the first two standard basis vectors that work are e, and ej then enter the values of i and j into the answer box below, separated with a comma. Problem #9:arrow_forwardGiven x=(1,-2,-1,a) and y=(3,3,-2,1) are two 1x4 row vectors, where "a" is some constant, which of the following statements is false? O y-2x = (1,7,0,-2a) x· 2y =a – 1 x and y are orthogonal if a=1. Ilx||=/6+a?arrow_forward
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- The following question is from linear algebra first year: Factors the vector (6, -5, -1)t into three components a,b,c that satisfy the following conditions: a depends on (2,0,1)t, b depends on (1,2, 0)t and c is orthogonal to a and b. Please show it step by step. Can we get integers as answers?arrow_forwardIf k is a real number, then the vectors (1,k),(k,7k+8) are linearly independent precisely when k ≠ a,b, where a = ? and b = ?, and a<barrow_forwardProblem 6. Prove that the vectors v₁ = (1, 0, -1), U₂ = (1, 2, 1), V3 = (0, -3,2) form a basis for R³. Express each of the standard basis vectors, (1, 0, 0), (0, 1, 0), and (0,0,1) as a linear combination of v₁, U2, U3.arrow_forward
- 1. Write the vector H- 16 as a linear combination of the vectors and Darrow_forwardThe following question is from linear algebra : Factors the vector (6, -5, -1)t into three components a,b,c that satisfy the following conditions: a depends on (2,0,1)t, b depends on (1,2, 0)t and c is orthogonal to a and b. Please show it step by step.arrow_forwardProblem #9: Given three linearly independent vectors V₁ = (-9,3,-9,9,-2), V2 = (-8, 10, 9, -3, 10), V3 = (-7,-9,-7, -7,0), in RS, complete this to a basis using the first two standard basis vectors which work. If the first two standard basis vectors that work are e, and e; then enter the values of i and j into the answer box below, separated with a comma. Problem #9:arrow_forward
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