p;lz provide answer for q27 asap Question 27Which of the following sets of vectors in M22 . the vector space of 2 x 2 matrices with real number entries, usual additionand scalar multiplication, are orthonormal with respect to the inner product (A, B) = trace(B" AY?Select ALL that apply.口「1o [1/2 1/2]-1/2-1/211/2 1/21/21/2]

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Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.6: Matrices

Problem 18E: Prove part b of Theorem 1.35. Theorem 1.35 Special Properties of Let be an arbitrary matrix...

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Find a basis for R3 that includes the vector (1,0,2) and (0,1,1).
Illustrate properties 110 of Theorem 4.2 for u=(2,1,3,6), v=(1,4,0,1), w=(3,0,2,0), c=5, and d=2. THEOREM 4.2Properties of Vector Addition and Scalar Multiplication in Rn. Let u,v, and w be vectors in Rn, and let c and d be scalars. 1. u+v is vector in Rn. Closure under addition 2. u+v=v+u Commutative property of addition 3. (u+v)+w=u+(v+w) Associative property of addition 4. u+0=u Additive identity property 5. u+(u)=0 Additive inverse property 6. cu is a vector in Rn. Closure under scalar multiplication 7. c(u+v)=cu+cv Distributive property 8. (c+d)u=cu+du Distributive property 9. c(du)=(cd)u Associative property of multiplication 10. 1(u)=u Multiplicative identity property
Prove part b of Theorem 1.35. Theorem 1.35 Special Properties of Let be an arbitrary matrix over. With as defined in the preceding paragraph,
Complete Example 2 by verifying that {1,x,x2,x3} is an orthonormal basis for P3 with the inner product p,q=a0b0+a1b1+a2b2+a3b3. An Orthonormal basis for P3. In P3, with the inner product p,q=a0b0+a1b1+a2b2+a3b3 The standard basis B={1,x,x2,x3} is orthonormal. The verification of this is left as an exercise See Exercise 17..

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Which of the following sets of vectors in M22, the vector space of 2 x 2 matrices with real number entries, usual addition and scalar multiplication, are orthonormal with respect to the Inner product (A, B) trace(B" A)? Select ALL that apply.