In this exercise, we are given a matrix $A$ and a vector $b$, with the matrix $A$ having three column vectors $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3$, and $W = \text{Span} \{ \mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3 \}$. Let's break down each part of the question.

1. Is ( \mathbf{b} \in {\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3}? How many vectors are in this set?

   • The set $\{\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\}$ consists of exactly three vectors. Therefore, the number of vectors in the set is 3.

2. Is $\mathbf{b} \in W$? How many vectors are in $W$?

   • $W$ is the span of $\{\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\}$, meaning it consists of all linear combinations of the vectors $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3$. The vector $\mathbf{b}$ can be expressed as a linear combination of $\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3$ as follows: $\mathbf{b} = (-4)\mathbf{a}_1 + (-1)\mathbf{a}_2 + (-2)\mathbf{a}_3$ Hence, $\mathbf{b} \in W$, and there are infinitely many vectors in $W$ because the span of three independent vectors is a subspace with infinitely many vectors.

3. Show that $\mathbf{a}_2$ is in $W$.

   • The vector $\mathbf{a}_2$ is one of the generators of $W$, meaning it is already part of the set that spans $W$. Therefore, $\mathbf{a}_2 \in W$ because $W = \text{Span} \{\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\}$, and any vector in the set of generators automatically belongs to $W$.

Let me know if you'd like further clarification on any of these points!

Related Questions:

1. What does it mean for a set of vectors to span a space?
2. How can you determine if a vector is in the span of other vectors?
3. What is the geometric interpretation of the span of vectors?
4. How can we verify linear independence of vectors in a span?
5. How do row operations help in determining whether a vector is in the span of other vectors?

Tip: When dealing with subspaces like spans, it's useful to remember that any vector in a subspace can be written as a linear combination of the generators (basis vectors) of that subspace.