Identify the correct alternative that answers the questions below, based on the given text: 7. Determine if the vectors are LD or LI, and if they ...

Identify the correct alternative that answers the questions below, based on the given text:

7. Determine if the vectors are LD or LI, and if they are LI, determine the dimension of S.

a) LD and 3.
b) LD and 2.
c) LI and 2.
d) LI and 1.
e) LI and 3.

8. Calculate the value detA - detB.

a) 1
b) -5
c) 0
d) -2
e) 3

9. Calculate the value (detA x detB).

a) 1
b) -5
c) -3
d) -1
e) 3

10. Is T(x,y,z) = (x + 2z , z – x , x + y + 2z ) a linear transformation? What is the associated linear transformation matrix for 'T'? What is the characteristic polynomial?

a) T is linear, [-1 0 2; 0 -1 1; 1 1 2], -X³+3X²+ X-3
b) T is not linear, [-1 0 2; 0 -1 1; 1 1 2], -X³+2X²+ X-2
c) T is linear, no characteristic polynomial.
d) T is linear, does not present a transformation matrix, -X³+2X²+ X-2
e) T is not linear, does not present a transformation matrix, no characteristic polynomial.

7. Determine if the vectors are LD or LI, and if they are LI, determine the dimension of S.
8. Calculate the value detA - detB.
9. Calculate the value (detA x detB).
10. Is T(x,y,z) = (x + 2z , z – x , x + y + 2z ) a linear transformation? What is the associated linear transformation matrix for 'T'? What is the characteristic polynomial?
a) LD and 3.
b) LD and 2.
c) LI and 2.
d) LI and 1.
e) LI and 3.
a) T is linear, [-1 0 2; 0 -1 1; 1 1 2], -X³+3X²+ X-3
b) T is not linear, [-1 0 2; 0 -1 1; 1 1 2], -X³+2X²+ X-2
c) T is linear, no characteristic polynomial.
d) T is linear, does not present a transformation matrix, -X³+2X²+ X-2
e) T is not linear, does not present a transformation matrix, no characteristic polynomial.