Determine if b is a linear combination of a1 a2, and a3. a1 = [ 1 - 2 0 ], a2 = [ 0 1 3 ], a3 = [ 6 - 6 18 ], b = [ 2 - 2 6 ] Choose the correct answer below. Vector b is a linear combination of a1 a2, and a3. The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the fourth column. Vector b is a linear combination of a1 a2, and a3. The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the third column. Vector b is not a linear combination of a1, a2, and a3. Vector b is a linear combination of a1 a2, and a3. The pivots in the corresponding echelon matrix are in the first entry in the first column and the second entry in the second column.

Question

Mathematics

Kathy

10 October, 00:09

Determine if b is a linear combination of a1 a2, and a3. a1 = [ 1 - 2 0 ], a2 = [ 0 1 3 ], a3 = [ 6 - 6 18 ], b = [ 2 - 2 6 ] Choose the correct answer below. Vector b is a linear combination of a1 a2, and a3. The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the fourth column. Vector b is a linear combination of a1 a2, and a3. The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the third column. Vector b is not a linear combination of a1, a2, and a3. Vector b is a linear combination of a1 a2, and a3. The pivots in the corresponding echelon matrix are in the first entry in the first column and the second entry in the second column.

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Answers (1)

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Israel White · Answer 1

Answer: Vector b is not a linear combination

Step-by-step explanation:

First of all we put the vectors in terms of different variables, such as:

a1 (1,-2,0) = (a,-2a, 0);

a2 (0,1,3) = (0, b, 3b);

a3 (6,-6,18) = (6c,-6c, 18c);

To know that a vector is a linear combination we need to express it like a sum of other different vectors.

(2,-2,6) = (a,-2a, 0) + (0, b, 3b) + (6c,-6c, 18c)

(2,-2,6) = (a+0+6c,-2a+b-6c, 0+3b+18c)

We express this sum like a system of equations.

a+6c=2

-2a+b-6c=-2

3b+18c=6

We solve this system of equations and we can note that the system don't have a solution, so the vector b is not a linear combination of a1, a2, and a3.