Let V=M(R) denote the space of 2x2 matrices over R. Define linear functionals h1, h2, h3,...

    

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Let V=M(R) denote the space of 2x2 matrices over R. Define linear functionals h1, h2, h3, h4 V R. h((b))=a+b-d = +c-2d nz ((ab)). =a+c-2d h3 = ((a b)) = = b+2d h4 4+ ((a b)) = c + 2d (a) Prove that C = {h1, h2, h3, h4] is a basis of the dual space V", and find a basis C of V such that C is dual to C. (7 marks) (b) Let B = ( ) B(23). B = (, 2). B3 and let W = span{B1, B2, B3}. Find an element f V* of the form f=ch + ch+ C3h3 + C4h4 such that W span{f} and C1, C2, C3, C4 are integers. Let V=M(R) denote the space of 2x2 matrices over R. Define linear functionals h1, h2, h3, h4 V R. h((b))=a+b-d = +c-2d nz ((ab)). =a+c-2d h3 = ((a b)) = = b+2d h4 4+ ((a b)) = c + 2d (a) Prove that C = {h1, h2, h3, h4] is a basis of the dual space V", and find a basis C of V such that C is dual to C. (7 marks) (b) Let B = ( ) B(23). B = (, 2). B3 and let W = span{B1, B2, B3}. Find an element f V* of the form f=ch + ch+ C3h3 + C4h4 such that W span{f} and C1, C2, C3, C4 are integers.