Q1: What is the correct definition for linear independence of a family $(\mathbf{u}, \mathbf{v}, \mathbf{w})$ of vectors in $\mathbb{R}^n$?

A1: If $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ is a linear combination that results in the zero vector, then we have $\lambda_1 = \lambda_2 = \lambda_3 = 0$.

A2: If $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ is a linear combination that results in the zero vector, then we have $\lambda_1 = \lambda_2 = \lambda_3 \neq 0$.

A3: If $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ is a linear combination for another vector, then we have $\lambda_1 = \lambda_2 = \lambda_3 \neq 0$.

A4: If $\lambda_1 = \lambda_2 = \lambda_3 \neq 0$, then $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ is a linear combination that results in the zero vector.

A5: If $\lambda_1 = \lambda_2 = \lambda_3 = 0$, then $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ is a linear combination that results in the zero vector.

Q2: What is the correct definition for linear dependence of a family $(\mathbf{u}, \mathbf{v}, \mathbf{w})$ of vectors in $\mathbb{R}^n$?

A1: There is a linear combination $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ that results in the zero vector, where $\lambda_1, \lambda_2$ and $\lambda_3$ are not all zero.

A2: If $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ is a linear combination that results in the zero vector, then we have $\lambda_1 = \lambda_2 = \lambda_3 \neq 0$.

A3: There is a linear combination $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ that results in the zero vector, where we have $\lambda_1 = \lambda_2 = \lambda_3 = 0$.

A4: There is a linear combination $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ that results in the zero vector, where we have $\lambda_1 = \lambda_2 = \lambda_3 \neq 0$.

A5: If $\lambda_1 = \lambda_2 = \lambda_3 = 0$, then $\lambda_1 \mathbf{u} + \lambda_2 \mathbf{v} + \lambda_3 \mathbf{w}$ is a linear combination that results in the zero vector.