We are given the following vectors:

• $w = 15x - 5y - 3z$
• $Z = 3x - y$

We are tasked with determining if the span of $\{x, y, z\}$ is equal to the span of $\{w, x, z\}$. To do this, we'll check if $y$ can be expressed as a linear combination of $w, x, z$, because if it can, then the span of $\{x, y, z\}$ is indeed equal to the span of $\{w, x, z\}$.

Step 1: Express $y$ in terms of $w, x, z$

Start with the expression for $w$:

$w = 15x - 5y - 3z$

Solve for $y$:

$w = 15x - 5y - 3z$ $w + 5y = 15x - 3z$ $5y = 15x - 3z - w$ $y = 3x - \frac{3}{5}z - \frac{1}{5}w$

This shows that $y$ can indeed be written as a linear combination of $w, x, z$.

Step 2: Conclusion

Since $y$ can be expressed as a combination of $w, x, z$, it follows that the span of $\{x, y, z\}$ is the same as the span of $\{w, x, z\}$. Therefore:

$\text{span}(x, y, z) = \text{span}(w, x, z)$

Would you like further details or have any questions?

Relative Questions:

1. What is the geometric interpretation of vector spans?
2. How do linear combinations relate to the span of a set of vectors?
3. Can the span of more than three vectors ever equal the span of three vectors?
4. How does the concept of span apply in higher dimensions, such as $\mathbb{R}^4$?
5. How can we generalize the relationship between two spans like in this problem?

Tip:

When checking if two spans are equal, express the vectors of one span in terms of the other. If all vectors in one span can be written as combinations of the other set, the spans are equal.