Your homework consists of six problems primarily focused on linear algebra, including topics such as linear dependence, span, vector operations, and solving linear systems. Here’s a brief overview of each problem:

1. Linear Combination Problem: Determine for which values of $h$ the vector $\mathbf{b}$ is a linear combination of the vectors $\mathbf{a}_1$ and $\mathbf{a}_2$.
2. Linear Independence and Span: Analyze whether the vectors $\mathbf{u}$ and $\mathbf{v}$ are linearly independent or dependent, list vectors in their span, and provide a geometric interpretation.
3. Linear Dependence of a Set: Demonstrate the linear dependence of a set of three vectors by finding a dependence relation.
4. Mining Company Output: Solve a system of equations involving copper and silver production to determine how long each mine should operate.
5. Solving a Linear System: Determine conditions for a linear system to have no solution, a unique solution, or infinitely many solutions.
6. Span and Vector Plotting: Analyze the span of two vectors, express linear combinations, and plot the geometric objects using a computer algebra system.

Do you need detailed solutions or guidance on any specific problem? Here are five relevant questions that may expand your understanding:

1. What is the significance of the span of two vectors in $\mathbb{R}^2$ or $\mathbb{R}^3$?
2. How do you find a dependence relation for a set of vectors?
3. What conditions on the constants $h$ and $k$ ensure that a linear system has no solution?
4. How does linear independence differ from linear dependence geometrically?
5. What method can you use to solve a system of vector equations?

Tip: When working with linear dependence, remember that if a set of vectors includes the zero vector or if one vector can be expressed as a linear combination of the others, the set is linearly dependent.