To find the vector $w$ such that $B = \{[1, 1, 1], [1, 0, 1], w\}$ forms a basis for $\mathbb{R}^3$, we can use the information given about the coordinate vector of $[-2, -7, 4]$ in this basis.

Given:

• $[1, 1, 1]$ and $[1, 0, 1]$ are known basis vectors.
• The coordinate vector of $[-2, -7, 4]$ in this basis is $[-1, 2, -3]$.

Using the coordinate vector, we can express $[-2, -7, 4]$ in terms of the basis $B$: [ [-2, -7, 4] = -1 \cdot [1, 1, 1] + 2 \cdot [1, 0, 1] + (-3) \cdot w ]

Expanding this equation: [ [-2, -7, 4] = [-1, -1, -1] + [2, 0, 2] - 3w ]

Combining terms on the right-hand side: [ [-2, -7, 4] = [1, -1, 1] - 3w ]

Now, we can isolate $w$ by rearranging terms: [ 3w = [1, -1, 1] - [-2, -7, 4] ]

Calculating the right-hand side: $3w = [1 + 2, -1 + 7, 1 - 4]$ $3w = [3, 6, -3]$

Now, divide by 3 to solve for $w$: $w = [1, 2, -1]$

Thus, the vector $w$ that completes the basis $B$ is: $w = [1, 2, -1]$

Would you like further details on this solution?

---

Additional Questions to Explore:

1. What are the properties of a basis in $\mathbb{R}^3$?
2. How do coordinate vectors relate to basis transformations?
3. Can you explain why a basis in $\mathbb{R}^3$ must consist of three vectors?
4. How would the solution change if the coordinate vector of $[-2, -7, 4]$ were different?
5. Why is it essential for the basis vectors to be linearly independent?

Tip: When working with coordinate vectors in a new basis, remember they represent a linear combination of basis vectors to express a vector in the standard coordinate system.