these are 2 seperate questions 1 2 7 5 11 8 what is the comp

these are 2 seperate questions

1 2 7 5 11 8, what is the complete solution of -1 rs -1 2 4 15 12 1 J 9. Determine whether the following set is a subspace of 3N3.

Solution

8.   Let A=

1

2

7

5

1

2

-1

-1

-5

-6

0

-1

2

4

15

12

1

3

We will reduce A to its RREF as under:

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

Add -2 times the 3rd row to the 2nd row ; Add -7 times the 3rd row to the 1st row

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

Then the RREF of A is

1

0

0

1

2

3

0

1

0

-5

3

3

0

0

1

2

-1

-1

Then the given equation is equivalent to x₁+x₄+2x₅ =3; x₂-5x₄+3x₅= 3 and x₃+2x₄ –x₅= -1. Now, let x₄ = r, and x₅ = t.Then x₁= 3-r-2t , x₂= 3+5r-3t and x₃=-1-2r +t so that (x₁,x₂,x₃,x₄,x₅)^T = (3-r-2t , 3+5r-3t, -1-2r +t, r,t)^T = ( 3,3,-1,0,0)^T +r(-1,5,-2,1,0)^T +t(-2,-3,1,0,1)^T, where r, t are arbitrary real numbers.

9. We have x₁ -2x₂ = 0 so that x₁ = 2x₂ and (x₁,x₂,x₃)^T = (2x₂,x₂,x₃)^T.

The conditions that a vector space must satisfy are as under:

1. For all X, Y , X+Y = Y+X ( commutativity of vector addition).     

2. For all X, Y, Z , (X+Y)+Z=X+(Y+Z) (Associativity of vector addition).

3. For all x, 0+X = X+0 = X ( Existence of Additive identity)

4. For any X, there exists a -X such that X+(-X)= 0 (Existence of additive inverse)

5. For all scalars r,s and vectors X, r(sX)=(rs)X( Associativity of scalar multiplication).

6. For all scalars r,s and vectors X , (r+s)X=rX+sX (Distributivity of scalaraddition).

7. For all scalars r and vectors X,Y, r(X+Y)=rX+rY (Distributivity of vector addition).

8. For all vectors X, 1X=X ( Existence of Scalar multiplication identity).

Apparently, the given set satisfies all the above conditions . The vector (0,0,0)^T is the zero vector and ( -x₂ ,-2x₂,-x₃)^T is the additive inverse of (2x₂,x₂,x₃)^T. Thus, the given set is a subspace of R³.

1	2	7	5	1	2
-1	-1	-5	-6	0	-1
2	4	15	12	1	3