Question

T F Any set of vectors containing the zero vector is linearly dependent

Answer 1

True, any set of vectors that contains the zero vector is linearly dependent because a nontrivial combination of the vectors (including the zero vector scaled by any scalar) can produce the null vector.

Step-by-step explanation:

True. Any set of vectors containing the zero vector is linearly dependent. This is because the null vector, which has all components equal to zero (0î + 0ç + 0k) and therefore no length or direction, can always be represented as a linear combination of other vectors in the set by simply scaling it by any scalar. In more technical terms, for a set of vectors that includes the zero vector, you can find a nontrivial combination (not all scalars being zero) such that when these scalars are applied to their respective vectors, their sum equals the zero vector. This is the very definition of linear dependency. For instance, if you consider any scalar 'a' not equal to zero, a∗0 + 0∗0 + ... + 0∗0 = 0, which demonstrates linear dependence because not all the scalars are zero (namely 'a' is not).

Linear independence would require that the only way to represent the zero vector as a linear combination of the set of vectors is to have all scalars that multiply each vector in the combination be zero. Since the set includes the zero vector itself, this condition is violated, and the set is dependent.

Answer 2

True

Step-by-step explanation:

Let B={
`v_1, v_2,\cdots,v_k,0`} a set of vectors containing the zero vector.

B is linear dependent if exist scalars
`a_0,a_1, a_2,\cdots,a_k` not all zero such that
`a_0*0+a_1*v_1+\cdots+a_k*v_k=0`.

Observe that if
`a_1=a_2=\cdots=a_k=0` and
`a_0=\lambda\\eq 0` then

`\lambda*0+0*v_1+\cdots+0*v_k=0`.

Then B is linear dependent.