Let us solve the problem step-by-step to determine whether $v$ is a linear combination of $v_1$ and $v_2$ in both cases.

Part (i): $v = (a, b)$, $v_1 = (1, 0)$, $v_2 = (1, 1)$ in $\mathbb{R}^2$

Linear combination condition:

We need to determine if there exist scalars $c_1$ and $c_2$ such that: $v = c_1 v_1 + c_2 v_2$ This expands to: $(a, b) = c_1(1, 0) + c_2(1, 1)$ Breaking this into components: a = c_1 + c_2 \tag{1} b = c_2 \tag{2}

Solving for $c_1$ and $c_2$:

From Equation (2): $c_2 = b$.

Substitute $c_2 = b$ into Equation (1): $a = c_1 + b$ $c_1 = a - b$

Conclusion:

Since we found explicit values for $c_1$ and $c_2$, the vector $v = (a, b)$ is a linear combination of $v_1$ and $v_2$. The scalars are: $c_1 = a - b, \quad c_2 = b$

Part (ii): $v = (0, 0, 1)$, $v_1 = (1, 0, 1)$, $v_2 = (0, 1, 1)$ in $\mathbb{R}^3$

Linear combination condition:

We need to determine if there exist scalars $c_1$ and $c_2$ such that: $v = c_1 v_1 + c_2 v_2$ This expands to: $(0, 0, 1) = c_1(1, 0, 1) + c_2(0, 1, 1)$ Breaking this into components: 0 = c_1 \tag{1 (x-component)} 0 = c_2 \tag{2 (y-component)} 1 = c_1 + c_2 \tag{3 (z-component)}

Solving for $c_1$ and $c_2$:

From Equation (1): $c_1 = 0$.

From Equation (2): $c_2 = 0$.

Substitute $c_1 = 0$ and $c_2 = 0$ into Equation (3): $1 = 0 + 0$ This is a contradiction.

Conclusion:

Since there is no solution to the system, the vector $v = (0, 0, 1)$ is not a linear combination of $v_1$ and $v_2$.

Final Results:

1. Case (i): $v = (a, b)$ is a linear combination of $v_1$ and $v_2$.
2. Case (ii): $v = (0, 0, 1)$ is not a linear combination of $v_1$ and $v_2$.

Would you like further clarification or another related example? Here are 5 related questions to expand your understanding:

1. What is the geometric interpretation of a vector being a linear combination of others?
2. How can you generalize this method to higher dimensions (e.g., $\mathbb{R}^4$)?
3. What is the role of the rank of the matrix formed by $v_1$ and $v_2$ in determining linear dependence?
4. Can you extend this example to check if $v$ is in the span of more than two vectors?
5. How can this problem be approached using matrix notation and row reduction?

Tip: Always verify your solution by substituting back into the original equations to ensure consistency.