To determine which sets of vectors are linearly dependent, let’s analyze each vector provided:

1. Vectors:

   • $u = \begin{bmatrix} -1 \\ 2 \end{bmatrix}$
   • $v = \begin{bmatrix} 2 \\ -4 \end{bmatrix}$
   • $w = \begin{bmatrix} -1 \\ -1 \end{bmatrix}$
   • $x = \begin{bmatrix} -1 \\ 1 \end{bmatrix}$
   • $y = \begin{bmatrix} 0 \\ 0 \end{bmatrix}$

2. Linear Dependence Criteria: Vectors are linearly dependent if there exists a non-trivial combination of them that results in the zero vector.

Checking each option:

• {u, w}: Check if $u$ and $w$ are linearly dependent.

  • Since $u \neq k \cdot w$ for any scalar $k$, they are linearly independent.

• {u, v}: Check if $u$ and $v$ are linearly dependent.

  • Notice that $v = -2 \cdot u$. This means $u$ and $v$ are linearly dependent.

• {x, y}: Check if $x$ and $y$ are linearly dependent.

  • Since $y$ is the zero vector, any set containing $y$ will be linearly dependent.

• {w, x}: Check if $w$ and $x$ are linearly dependent.

  • These two vectors do not have a scalar multiple relationship and are linearly independent.

• {u, w, x}: Check if $u$, $w$, and $x$ are linearly dependent.

  • The combination of these three vectors does not allow for a non-trivial solution leading to the zero vector, so they are linearly independent.

• {u}: A single vector by itself is linearly independent unless it is the zero vector, which $u$ is not.

Answer:

The sets of vectors that are linearly dependent are:

• {u, v}
• {x, y}

Would you like further details on any specific steps or concepts?

Follow-up Questions:

1. How can we determine if a larger set of vectors is linearly dependent?
2. What is the importance of the zero vector in determining linear dependence?
3. How do scalar multiples play a role in linear dependence between vectors?
4. Why are sets containing the zero vector always linearly dependent?
5. How can we use matrix methods to verify linear dependence?

Tip: When assessing linear dependence, always check for scalar multiples or combinations that yield the zero vector.