To determine the dimensions of the vector spaces spanned by the sets of vectors given in each statement, let’s analyze each set.

Problem Analysis

1. First Set of Vectors: Vectors: $\begin{bmatrix} 1 \\ 1 \end{bmatrix}, \quad \begin{bmatrix} -1 \\ 1 \end{bmatrix}, \quad \begin{bmatrix} -1 \\ 1 \end{bmatrix}$

   To determine the dimension, we need to check if these vectors are linearly independent:

   • Notice that the second and third vectors are identical, so they are linearly dependent.
   • Thus, the span of these vectors is essentially determined by just the first two distinct vectors: $\begin{bmatrix} 1 \\ 1 \end{bmatrix} \text{ and } \begin{bmatrix} -1 \\ 1 \end{bmatrix}$
   • Since these two vectors are linearly independent, they span a 2-dimensional space.

   Therefore, the dimension of the vector space spanned by these vectors is 2.

2. Second Set of Vectors: Vectors: $\begin{bmatrix} 1 \\ 1 \end{bmatrix}, \quad \begin{bmatrix} -1 \\ -1 \end{bmatrix}, \quad \begin{bmatrix} -2 \\ -2 \end{bmatrix}$

   Analyzing linear independence:

   • Each of these vectors is a scalar multiple of $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$.
   • Therefore, all three vectors are linearly dependent, and they span only a 1-dimensional space (along the direction of $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$).

   The dimension of the vector space spanned by these vectors is 1.

Conclusion

The second statement, stating that the dimension is 1, is correctly checked as true. The first statement, stating that the dimension is 2, is also correct.

Would you like further explanation on linear independence, or do you have any questions?

Relative Questions

1. What is the definition of linear independence for vectors?
2. How do scalar multiples affect the span of a set of vectors?
3. What is the rank of a matrix, and how does it relate to the dimension of the vector space?
4. How can we use row reduction to determine the dimension of a vector space?
5. What does it mean for vectors to span a vector space?

Tip

When analyzing the span of vectors, look for scalar multiples to identify dependencies quickly.