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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.

Your homework consists of six problems primarily focused on linear algebra, including topics such as linear dependence, span, vector operations, and solving linear systems.

This homework covers six linear algebra problems involving linear dependence, vector span, and solving linear systems. Learn about the conditions for linear combinations, identifying linearly independent sets, vector operations, and more. Suitable for undergraduate students taking Math 220.

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