(Topsider the matrices A and B and the vecwt I,A = XXXXXXXXXX —1.) 2 B= -2 I XXXXXXXXXX -- i I . b=...

Question

(Topsider the matrices A and B and the vecwt I,
A = XXXXXXXXXX —1.) 2 B= -2 I XXXXXXXXXX -- i I . b= XXXXXXXXXX •
Find the inverse of A and use it to solve the system of linear equations Ax = b. Solve the system of equations Bx = b. Express each solution in vector form (as x = p (list + • • • + rusk, k > 0). Then answer the following questions. (i) In one sentence, state what the rank of a matrix is in relation to the reduced row echelon form of the matrix. Write down the ranks of the matrices A and B. (ii) What is the null space, N(M), of any matrix M? For the matrices given above, what subspace of R" is N(A)? Write down a basis for the subspace N(B). (iii) Denote the column vectors of A by el ,e2,c3 and let S = Justify that S is basis of 1R3. Write down the coordinates of the vector b in this basis; that is find Ms. (iv) Denote the columns of B by v1, v2, v3, v4. Are the column vectors linearly independent? Justify your answer. Write down a non-trivial linear combination of v1, v2, va, v4 which is equal to the zero vector, 0.

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Robert · Accepted Answer

(ii) Null space: 
  The null space or kernel of any matrix M is defined as the set of all vectors such that,
Given, 
  (
    
   
   
) 
From our definition for null space,
Let,   [
  
  
  
] 
 The equation reduces to, 
(
    
   
   
) [
  
  
  
]
Using row transformations, the equation can be written as,

(Topsider the matrices A and B and the vecwt I, A = XXXXXXXXXX —1.) 2 B= -2 I XXXXXXXXXX -- i I . b= XXXXXXXXXX • Find the inverse of A and use it to solve the system of linear equations Ax = b. Solve...

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