To solve for the real values of $a$ and $b$ so that vector $w = (-a, -1, b)$ belongs to the linear span of $u = (3, -3, 1)$ and $v = (-1, 1, 1)$, we need to express $w$ as a linear combination of $u$ and $v$. In other words, we want to find scalars $\lambda$ and $\mu$ such that:

$w = \lambda u + \mu v$

This equation in terms of components is:

$(-a, -1, b) = \lambda (3, -3, 1) + \mu (-1, 1, 1)$

Breaking this into a system of equations for each component:

1. $-a = 3\lambda - \mu$
2. $-1 = -3\lambda + \mu$
3. $b = \lambda + \mu$

Step 1: Solve for $\mu$ in terms of $\lambda$

From the second equation:

$-1 = -3\lambda + \mu$ $\mu = -1 + 3\lambda$

Step 2: Substitute $\mu$ into the first and third equations

Using $\mu = -1 + 3\lambda$ in the first equation:

$-a = 3\lambda - (-1 + 3\lambda)$ $-a = 3\lambda + 1 - 3\lambda$ $-a = 1$ $a = -1$

Now, substitute $\mu = -1 + 3\lambda$ into the third equation:

$b = \lambda + (-1 + 3\lambda)$ $b = \4\lambda - 1$

Step 3: Conclusion

We have determined the following:

• $a = -1$
• $b = 4\lambda - 1$ (which means $b$ can take any value depending on $\lambda$).

Thus, $a = -1$, and $b$ can be any real number, depending on the choice of $\lambda$.

Would you like more details or have any questions?

Here are 5 related questions to extend this problem:

1. How can you geometrically interpret the condition that $w$ belongs to the span of $u$ and $v$?
2. How would the solution change if $w$ were required to be orthogonal to both $u$ and $v$?
3. What is the dimensionality of the span of vectors $u$ and $v$?
4. How would the system change if we added another vector, say $z = (1, 0, 0)$, to the span?
5. How can you verify if a given vector belongs to the span of multiple vectors?

**Tip