pLEASE solve and explain question 12 Find the value of k suc

pLEASE solve and explain question 12

Find the value of k such thet the vectors (3, -k, 0) (5,1, 7) are orthegonal, b. Find the set of S of all vectors orthegonal to both (-2, 5,1) and (-3, 4,1) given matrix A = [1 2 -1 1 2 4 -3 0 1 2 1 5] 13. a. Find the rank (A) b. Find the basis for the row space of A, and dimension of the row space. c. Find the basis for null (A) and dimension of null(A) d. Find the basis for the col (A^T) and the dimension of column space (A^T) e. Show thet null (A) (col A^T)

Solution

12. a) if two vectors are ortogonal, then their dot product is zero.

(3 , -k, 0) and ((5,1,7): 3*15 -k*1+0 =0

15 = k

b) set of all vectors orthogonal to both (-2.5,1) and (-3,4,1)

Let set of orthogonal vector be ( x, y, z)

So, -2x +5y+z =0 ;

-3x+4y+z =0

multiplying equ 1 by 3 and equation 2 by 2:

-6x +15y +3z

-6x +8y +2z =0

----------------------- (subtract)

7y +z=0

y = -z/7

-3x +4(-z/7) +z =0

3x = 3z/7 ----> x = z/7

So, set of vector:( z/7 , -z/7 , z) = z( 1/7 , -1/7 , 1) orthogonal to given vectors