Question

I have attached the questions and the final solutions. Please do all of questions 7 and 8

7. Given the line L:x,y,z-2,2,3+1,-1,-3, the plane S 3x-2y+2z-7 and the point A 1,1,1 a) Find parametric equations of the line which contains the point A, intersects the line and which is parallel to the plane b) Find parametric equations of the lne which contains the point A and which intersects the line Lat the&angle a) Show that V" is a subspace of M b) Find a basis and the dimension of V ANSWERS 1. a) (1+2t,-8t,-1-t,tteR b) (-2,8,1) 2. a) k 0vk-1 b) k#0^ k c) there is no such k 1 0 101 4b) 010 3. a) A 001 2 4. a) Hint: Use the definition of orthogonal matrix. b) Hint: Write adjA in terms of A1 6. a) Hint: Use Properties of transpose operation. b) 2 17 4 010 001 000 8. a) Hint: Use the theorem. b-10000001is the basis and dim Va 3 100 0-10 9. a) Hint: Use the theorem. The coordinates are 1,2,3. b) teR 1At-7 10. a) False b) False

Answer 1

Question 8,

$M_{n\times n} \textrm{ is a vector space under usual matrix addition and scalar multiplication.} \\ \textrm{ For any, } A,B \in V_n, A^T = - A, B^T=-B, \; \therefore (A+B)^T = A^T + B^T = - A - B = - (A+B), \; \therefore A+B \in V_n. \\ \textrm{For any scalar, } \alpha \in F, \textrm{ and } A \in V_n, (\alpha A)^T = \alpha A^T = \alpha(-A)= -\alpha A. \therefore \alpha A \in V_n. \textrm{ This shows that } V_n \textrm{ is a subspace of } M_{n\times n}$

$V_3 = \{A \in M_{3\times 3} : A^T = - A\} = \{ \left( {\begin{array}{ccc} 0 & a_{12} & a_{13} \\ -a_{12} & 0 & a_{23} \\ -a_{13} & -a_{32} & 0 \end{array}} \right) : a_{12}, a_{13}, a_{23} \in F \} \\ \textrm{This is clearly of dimension 3. } \\ \textrm{A basis can be taken as similar to the standard ones, that is by plugging 1 in one of the variables and 0 in others }$

This will give you the answer as given by you in the picture provided.