(a) The function T: R R is defined as follows: 3 0 2 1 -1 -2...

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(a) The function T: R² R³ is defined as follows: 3 0 2 1 -1 -2 i. Find T(7), where = (2,-1). ii. Show that T is a linear transformation from R² into R³. (b) Express the quadratic form as a product of matrices. (d) If A = T(u) = Au = (4 marks) (c) Write the matrix A below as the sum of a symmetric and skew symmetric matrix. 1 2 4 -2 5 3 ·(-23). (²2₂3) A(BC) and A(B+C) = AB + AC. 3 ii. 1 0 Q(T1, T2, T3) = x + 2x3 + 2x32x1x2 - 2x2x3 iii. 4 2 1 1 -1 2 3 1 2 3 -2 1-3 A = ,B= = V2 6 3/ (e) Expand the following determinants by two methods: 1. along the third row, 2. along the third column. 1-3 2 i. 4 -1 2 3 5 2 and C= (2 marks) (3 marks) (6 marks) (2³). Verify that (AB)C= (7 marks) (6 marks) (6 marks) (6 marks) (f) Show that any square matrix can be expressed as the sum of two matrices, one symmetric and the other anti-symmetric. (5 marks) (a) The function T: R² R³ is defined as follows: 3 0 2 1 -1 -2 i. Find T(7), where = (2,-1). ii. Show that T is a linear transformation from R² into R³. (b) Express the quadratic form as a product of matrices. (d) If A = T(u) = Au = (4 marks) (c) Write the matrix A below as the sum of a symmetric and skew symmetric matrix. 1 2 4 -2 5 3 ·(-23). (²2₂3) A(BC) and A(B+C) = AB + AC. 3 ii. 1 0 Q(T1, T2, T3) = x + 2x3 + 2x32x1x2 - 2x2x3 iii. 4 2 1 1 -1 2 3 1 2 3 -2 1-3 A = ,B= = V2 6 3/ (e) Expand the following determinants by two methods: 1. along the third row, 2. along the third column. 1-3 2 i. 4 -1 2 3 5 2 and C= (2 marks) (3 marks) (6 marks) (2³). Verify that (AB)C= (7 marks) (6 marks) (6 marks) (6 marks) (f) Show that any square matrix can be expressed as the sum of two matrices, one symmetric and the other anti-symmetric. (5 marks)