# Linearly independent/Simple properties/Fact

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Let be a field, let be a -vector space, and let , , be a family of vectors in . Then the following statements hold.

- If the family is linearly independent, then for each subset , also the family , , is linearly independent.
- The empty family is linearly independent.
- If the family contains the null vector, then it is not linearly independent.
- If a vector appears several times in the family, then the family is not linearly independent.
- A single vector is linearly independent if and only if .
- Two vectors and are linearly independent if and only if is not a scalar multiple of and vice versa.