c What is the dimension of the null space In other words how

(c) What is the dimension of the null space? In other words, how many linearly independent solutions you can find for the following system:

Solution

(c ) The coefficient matrix for the given system of linear equations is A =

1

3

1

1

2

-2

1

2

1

-5

0

1

We will reduce A to its RREF as under:

Add -2 times the 1st row to the 2nd row

Add -1 times the 1st row to the 3rd row

Multiply the 2nd row by -1/8

Add 8 times the 2nd row to the 3rd row

Add -3 times the 2nd row to the 1st row

Then the RREF of A is

1

0

5/8

1

0

1

1/8

0

0

0

0

0

Thus, the given system of linear equations is equivalent to x₁ +5x₃/8 = 0 and x₂ +x₃/8 = 0. Let x₃ = -8t. Then x₁ = 5t and x₂ = t so that the solutions to the equation Ax = 0 ( i.e. the matrix representation of the given system of linear equations) is x = (5t,t,-8t)^T = t (5,1,-8)^T. Thus, the given linear system has infinite solutions, but only one linearly independent solution i.e. (5,1,-8)^T. The basis of the null space of A is also {(5,1,- 8)^T}. The dimension of the null space of A is 1.

4. The characteristic equation of the given matrix A is det(A- I₂) = 0 or, ²   -5 +6 = 0 (-3)( -2)= 0. Thus, the eigenvalues of A are ₁ = 3 and ₂ = 2. The corresponding eigenvectors being solutions to the equation Ax = x are v₁ =(4,3)^T and v₂ = (1,1)^T.

The required matrix S has columns which are eigenvectors of A. Thus S =

4

1

3

1

Please post the remaining questions again.

1	3	1	1
2	-2	1	2
1	-5	0	1