How to determine whether T is linearly independent ? Justify your answer please.

Question

Let S={X1,X2,X3} be a linearly independent set of vectors in R^3. How to determine whether T={2X1 + X2 - 2X3,4X1+ 3X2,3X1 + 2X2 - X3} is linearly independent. Justify your answer please.

Answer 1

2x₁ + x₂ - 2x₃

4x₁ + 3x₂

3x₁ + 2x₂ - x₃

Now we we will use Gaussian Elimination:

[2 1 -2]

[4 3 0] (R₂ = -2R₁ + R₂)

[3 2 -1] (R₃ = -3R₁ + 2R₁)

[2 1 -2]

[0 1 4]

[0 1 4] (R₃ = R₂ - R₃)

[2 1 -2]

[0 1 4]

[0 0 0] This is now in row echelon form. There is a pivot in the first row (2) and there is a pivot in the second row (1). There is no pivot in the third row, however, because the bottom row of the matrix only contains zero's and thus the matrix is not in upper triangular form (google upper triangular matrix). Since there is not a pivot in every row, T is not linearly independent.