Is this set linearly dependent?

Question

Is $\{v_1, \overrightarrow{0} \}$ linearly dependent for $v_1 \in \mathbb{R}^n$.

We have $\overrightarrow{0} = 0 v_1$, but but isnt that the "trivial solution" so the set is indeed linearly independent?

Answer 1

Any set containing the zero vector is linearly dependent.

In $\{\vec v_1, \vec 0\}$, you can write $\vec 0 = 0\cdot \vec v_1+1 \cdot\vec 0$ so it's possible to write the zero vector as a linear combination of the vectors in the set, without having to take all the coefficients equal to $0$.

Answer 2

Any set that contains $\overrightarrow{0}$ can't be linearly independent, because you can do any linear combination $0v_1+a\overrightarrow{0}$ with $a\in \mathbb{R}$ to generate the $0$ vector.

Answer 3

No, this set is always linearly dependent. The reason why is we can take any nonzero scalar $a$ and write

$$ \vec{0} \;\; =\;\; 0\cdot v_1 + a \cdot \vec{0}. $$