Let's analyze each statement and determine whether it is true or false:

1. Statement 1: "If $\{\vec{x}, \vec{y}, \vec{z}\}$ is linearly independent, then $\{\vec{y}, \vec{z}\}$ is also linearly independent."

   • False. If a set of vectors is linearly independent, removing one vector from the set does not guarantee that the remaining vectors will be linearly independent. For instance, if $\vec{x}$ is not in the span of $\vec{y}$ and $\vec{z}$, then $\vec{y}$ and $\vec{z}$ could still be linearly dependent.

2. Statement 2: "If all of $\vec{x}_1, \vec{x}_2, \ldots, \vec{x}_k$ are nonzero, then $\{\vec{x}_1, \vec{x}_2, \ldots, \vec{x}_k\}$ is linearly independent."

   • False. Just because each vector is nonzero does not imply that they are linearly independent. For example, if one vector is a scalar multiple of another, the set is linearly dependent despite all vectors being nonzero.

3. Statement 3: "If $a\vec{x} + b\vec{y} + c\vec{z} = \vec{0}$, then $\{\vec{x}, \vec{y}, \vec{z}\}$ is linearly independent."

   • False. This equation implies that $\vec{x}, \vec{y}$, and $\vec{z}$ satisfy a linear combination equal to zero. If $a$, $b$, and $c$ are not all zero, then $\vec{x}$, $\vec{y}$, and $\vec{z}$ are linearly dependent. For them to be independent, the only solution would need to be $a = b = c = 0$.

4. Statement 4: "If $\{\vec{x}_1, \vec{x}_2, \ldots, \vec{x}_k\}$ is linearly dependent, then $t_1\vec{x}_1 + t_2\vec{x}_2 + \cdots + t_k\vec{x}_k = \vec{0}$ for some numbers $t_i$ in $\mathbb{R}$ not all zero."

   • True. By definition, if a set of vectors is linearly dependent, then there exists a nontrivial combination (not all coefficients zero) of these vectors that equals the zero vector.

Summary:

1. False
2. False
3. False
4. True

Would you like more details on any of these statements, or have any questions?

Here are some related questions to explore further:

1. What is the geometric interpretation of linear independence in $\mathbb{R}^3$?
2. How does linear independence relate to the rank of a matrix?
3. Can a set containing only two vectors be linearly dependent? Under what conditions?
4. How does linear dependence change if we add a vector that is a linear combination of existing vectors?
5. Why is the zero vector always linearly dependent with any other set of vectors?

Tip: When verifying linear independence, check if the only solution to the vector equation is the trivial one (all scalars equal zero).