NYJC 2026 FM TP - Linear Algebra Set 3

2026 J2 H2 FM - Linear Algebra (Set 3) 1 In this question, V denotes the set of vectors of the form a b c d       , where a, b, c and d are real numbers. You may assume that V forms a linear space under the usual operations of vector addition and multiplication by a scalar. (a) Show that the subset of V for which 2a b d c+ − = does not form a linear space. [1] (b) Determine whether or not the subset of V for which both 42a b c d+ = + and 2 a b c d− = + forms a linear space. [2] (c) Determine whether or not the subset of V for which 4 2 a b c d+ = + or 2a b c d− = + forms a linear space. [2] (d) State the dimension of the subspace of V such that 3 2 0 a b c d+ − + = and provide a basis for this subspace. [3] 2 An orthogonal matrix Q is one such that T ,=Q Q I where I is the identity matrix. (a) Find the possible determinants of an orthogonal matrix. [3] For any vectors 3,, uv  it can be shown that T .=u v u v (b) Suppose 1 2 3, andv v v are the column vectors (in that order) of a 3  3 orthogonal matrix Q with real entries. Use the above result to show that 1 2 3, andv v v are unit vectors that are pairwise perpendicular. [4] Suppose 1 2 3 1 2 2 1 1 12 , 1 and 2 ,3 3 32 2 1 −           = = − = −                v v v and the linear transformation T is defined as follows: ( ) 1 2 3 T     = u if and only if 1 1 2 2 3 3 ,  = + +u v v v where 1 2 3, and    are real constants. (c) The column vectors of an orthogonal matrix are called orthonormal vectors. Given that 12,vv and 3v as defined above are orthonormal vectors, find the matrix A representing the linear transformation T. [2] (d) Hence find the subspace of 3  whose range space under T is given by : , . p q p q pq    +  [3]

3 Let V be the vector space of all real polynomials of degree at most n, and let :D V V→ be the linear transformation representing differentiation, that is, d() d pDp x= . (a) Explain why D is a linear transformation. [1] (b) State bases for the kernel and range of D. [2] (c) Give a reason why the set of real polynomials of degree n is not a vector space. [1] 4 Do not use a calculator in answering this question. The linear transformation T, from 22 ,→  defined by T : where , where , , and are constants .x x a b a b c dy y c d       =           AA  It is given that the transformation under T has a line 1L of invariant points. (a) Explain why A has an eigenvalue equal to 1. [1] (b) Hence show that 1ad a d bc+ = + + . [2] It is given that 341 435 =  − A . (c) Find an eigenvector 1e that corresponds to eigenvalue 1 of A and the Cartesian equation of 1.L [2] It is given that the transformation under T has an invariant line 2.L (d) By finding the other eigenvalue and its corresponding eigenvector 2e of ,A give the Cartesian equation of 2.L [3] (e) Show that 1L and 2L are perpendicular. [1] (f) By considering your answer to part (c), (d), (e) and the image of  +12ee when it is transformed under T, where ,,   describe the geometrical transformation T explaining your answer. [3]