Three n−vectors  and u v w are linear independent. Show that u, (u+v), (u+v+w) are linear independent.

To prove that the vectors \( \mathbf{u} \), \( \mathbf{u} + \mathbf{v} \), and \( \mathbf{u} + \mathbf{v} + \mathbf{w} \) are linearly independent, given that \( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{w} \) are linearly independent, we need to show that the only solution to the following linear combination being equal to the zero vector is when all coefficients are zero:

\[
c_1\mathbf{u} + c_2(\mathbf{u} + \mathbf{v}) + c_3(\mathbf{u} + \mathbf{v} + \mathbf{w}) = \mathbf{0}
\]

Let's expand this expression:

\[
c_1\mathbf{u} + c_2(\mathbf{u} + \mathbf{v}) + c_3(\mathbf{u} + \mathbf{v} + \mathbf{w}) = \mathbf{0}
\]

Distribute \( c_2 \) and \( c_3 \):

\[
c_1\mathbf{u} + c_2\mathbf{u} + c_2\mathbf{v} + c_3\mathbf{u} + c_3\mathbf{v} + c_3\mathbf{w} = \mathbf{0}
\]

Now, combine like terms:

\[
(c_1 + c_2 + c_3)\mathbf{u} + (c_2 + c_3)\mathbf{v} + c_3\mathbf{w} = \mathbf{0}
\]

Since \( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{w} \) are linearly independent, the coefficients of \( \mathbf{u} \), \( \mathbf{v} \), and \( \mathbf{w} \) must each be zero. This gives us the following system of equations:

\[
\begin{aligned}
c_1 + c_2 + c_3 &= 0 \quad \text{(1)} \\
c_2 + c_3 &= 0 \quad \text{(2)} \\
c_3 &= 0 \quad \text{(3)}
\end{aligned}
\]

### Step 1: Solve for \( c_3 \)
From equation (3), \( c_3 = 0 \).

### Step 2: Solve for \( c_2 \)
Substitute \( c_3 = 0 \) into equation (2):

\[
c_2 + 0 = 0 \implies c_2 = 0
\]

### Step 3: Solve for \( c_1 \)
Substitute \( c_2 = 0 \) and \( c_3 = 0 \) into equation (1):

\[
c_1 + 0 + 0 = 0 \implies c_1 = 0
\]

### Conclusion
Since \( c_1 = 0 \), \( c_2 = 0 \), and \( c_3 = 0 \) are the only solutions to the linear combination being zero, the vectors \( \mathbf{u} \), \( \mathbf{u} + \mathbf{v} \), and \( \mathbf{u} + \mathbf{v} + \mathbf{w} \) are linearly independent.

Would you like any further clarification on this proof or have additional questions?

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Here are some related questions:

1. What does it mean for vectors to be linearly independent?
2. How can you determine if a set of vectors is linearly independent in general?
3. Can linearly independent vectors become linearly dependent if one vector is removed?
4. What is the geometric interpretation of linear independence in 2D and 3D spaces?
5. How does the concept of linear independence relate to the rank of a matrix?
6. Can vectors be linearly independent if they are not in the same dimension?
7. What is the relationship between linear independence and the determinant of a matrix?
8. How does linear independence relate to the solution of a system of linear equations?

**Tip:** When dealing with questions of linear independence, always check if the only solution to a combination equating to zero is when all coefficients are zero. This is the defining property of linear independence.

Learn how to prove the linear independence of vectors u, u+v, and u+v+w given that u, v, and w are linearly independent. This proof involves demonstrating that the only solution to a specific linear combination equating to zero is when all coefficients are zero.

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