Which of these vectors can be written as a linear combinatio

Which of these vectors can be written as a linear combination of [6,6,4,4] and [6,4,6,2]?

A) [120,17,27,40]

B) [60,54,10,34]

C) [45,26,75,46]

D) [12,14,18,10]

Choose all that apply.

Solution

We form 3 x 4 matrices with the given vector as the 1st and the 2nd rows. The 3rd row will be, one by one, from the given choices. Then the 3 X 4 matrix so formed is reduced to its RREF by row operations.

A). The RREF has all non-zero rows. Therefore, the vector [ -60, 54, 10,34] cannot be expressed as a linear combination of the given vectors.

B) The 3rd row of the RREF is a zero row. Therefore, the vector [ -60, 54, 10,34] can be written as a linear combination of the given vectors.

C) he RREF has all non-zero rows. Therefore, the vector [ -45, 26, 75,46] cannot be expressed as a linear combination of the given vectors.

D) The 3rd row of the RREF is a zero row. Therefore, the vector [ -12, 14, 18,10] can be written as a linear combination of the given vectors.