Question

Are the following vectors linearly independent?

1. (1, 2), (-2, 4)
2. (0, 0, 1), (0, 1, 1), (1, 1, 1)
3. (1, 2), (1, 3), (1, 1)
4. (1, 2, 3), (2, 3, 4)
5. (0, 1, 2), (2, 0, 1), (0, 0, 1), (3, 2, 1)

Answer 1

1. Since $\frac{1}{2}\ne\frac{-2}{4},$ the vectors $(1, 2), (-2, 4)$ are not collinear, and thus they are linearly independent.

2. Consider the linear combination $a(0, 0, 1)+b (0, 1, 1)+c (1, 1, 1)=(0,0,0),$ which is equivalent to $(c,b+c,a+b+c)=(0,0,0),$ and hence $a=b=c=0.$ We conclude that the vectors $(0, 0, 1), (0, 1, 1), (1, 1, 1)$ are linear independent.

3. Since in 2-dimentinal space any three vectors are linearly dependent, we conclude that the vectors $(1, 2), (1, 3), (1, 1)$ are linearly dependent.

4. Since $\frac{1}{2}\ne\frac{2}{3}\ne\frac{3}4,$ we conclude that the vectors$(1, 2, 3), (2, 3, 4)$ are not collinear, and hence they are linearly independent.

5. Since in 3-dimentinal space any four vectors are linearly dependent, we conclude that the vectors $(0, 1, 2), (2, 0, 1), (0, 0, 1), (3, 2, 1)$ are linearly dependent.