NJ Topical Revision - Linear Algebra

National Junior College Mathematics Department H2 FMaths / Revision Materials by Topics Page 72 of 103 F07: Matrices and Linear Spaces 1 (a) A square matrix A is called orthogonal if and .T T AA I A A I Show that if A and B are orthogonal matrices, then AB is an orthogonal matrix. [2] (b) Let 1u , 2u , … , 1ku , ku be vectors in a vector space V such that  1 1,span , k V u u . Show that 1u , 2u , … , 1ku , ku are linearly dependent. [2] (c) (i) Let A be a square matrix and  is an eigenvalue of A with corresponding eigenvector u . Show that 2 is an ei genvalue of 2A with correspon ding eigenvector u . [2] (ii) Suppose further that 2 A A . Show that if  is an eigenvalue of A , then 0  or 1. [2] (2016 NYJC / JC1 / MYE / Q5) 2 It is given that the eigenvalues 1 , 2 , 3 of the matrix 3 1 1 8 8 8 1 1 1 4 2 4 1 1 5 8 8 8 a a a                   M , are the roots of the equation 3 2) )32( 48( 2 0.(2 ) 3a a a        Find 1 , 2 , 3 in terms of a. [3] Find matrices Q and D such that 1M QDQ , where the elements of Q are independent of a, and D is a diagonal matrix. [The evaluation of 1Q is not required. [5] Find the set of values of a such that all the elements of nM tend to zero as n . [3] (2016 NYJC / JC1 / MYE / Q6)

National Junior College Mathematics Department H2 FMaths / Revision Materials by Topics Page 73 of 103 3 A linear transformation 3 3:L   is defined by 2 , x x z L y x y z y az                    where a is a real constant. (a) Determine the conditions for a and b such that the point (2, 3, ) b is in the range of L . [3] (b) Find the value of a such that the null space of L is non-trivial. Find a basis for the range of L and a basis for the null space of L for this value of a. [5] For this value of a, (i) show that (0, 0, 0) is the only invariant point under L , [2] (ii) find, in the form  r a b , an equation for the line whose image under L is the point with position vector 5 3 2         , [3] (iii) show that the image under L of the plane 2x y z   is the plane 0x y z   . [3] (2016 NYJC / JC1 / MYE / Q7) 4 Let nP be the set of polynomials with real coefficients of the form 2 0 1 2( ) n np x a a x a x a x      . (i) Assume that nP is a vector space over  , write down the standard basis for thi s vector space. [1] (ii) Let { ( ) : (0) 0}nV p x P p   . Show that V is a subspace of nP . [3] Every polynomial with real coefficients of the form 2 0 1 2 n na a x a x a x     can be expressed as a ( 1) 1n  column vector 0 1 n a a a              . The I -transformation, which acts on elements of nP , is defined as 0 ( ) ( ) x I p p t dt . (iii) Show that 1: n nI P P  is a linear transformation. Write down the matrix representing I and find a basis for the range spac