being vector u (3, -3, 1), v (-1, 1, 1) and w (-a, -1, b). Determine all the real values of a and b so that: vector w belons to L{(u, v)}

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

Being vector u (3, -3, 1), v (-1, 1, 1) and w (-a, -1, b), determine all the real values of a and b so that vector w belongs to L{(u, v)}.

This problem explores how to determine the real values of a and b so that the vector w (-a, -1, b) lies in the span of two given vectors u (3, -3, 1) and v (-1, 1, 1). It covers key linear algebra concepts like linear combinations and vector spans, suitable for undergraduate-level students.

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