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do questions 3 4 5 and 6 make sure to only use matrices and use gaussian elimnation

### Problem 3: 
We are given the set of vectors:
\[
\left\{ \begin{bmatrix} 1 \\ 0 \\ -3 \end{bmatrix}, \begin{bmatrix} 2 \\ 5 \\ 1 \end{bmatrix}, \begin{bmatrix} -4 \\ -15 \\ -9 \end{bmatrix} \right\}
\]
We need to show that this set is linearly dependent by finding a dependence relation.

To solve this, let the vectors be \(v_1 = \begin{bmatrix} 1 \\ 0 \\ -3 \end{bmatrix}\), \(v_2 = \begin{bmatrix} 2 \\ 5 \\ 1 \end{bmatrix}\), and \(v_3 = \begin{bmatrix} -4 \\ -15 \\ -9 \end{bmatrix}\). We need to find scalars \(c_1, c_2, c_3\) such that:
\[
c_1 v_1 + c_2 v_2 + c_3 v_3 = 0
\]
This gives the system of equations:
\[
\begin{aligned}
c_1 + 2c_2 - 4c_3 &= 0 \\
5c_2 - 15c_3 &= 0 \\
-3c_1 + c_2 - 9c_3 &= 0
\end{aligned}
\]
We can solve this system using Gaussian elimination to find the values of \(c_1, c_2, c_3\).

### Problem 4:
We are given two vectors representing the daily output of two mines:
\[
v_1 = \begin{bmatrix} 20 \\ 550 \end{bmatrix}, \quad v_2 = \begin{bmatrix} 30 \\ 500 \end{bmatrix}
\]
#### Part (a):
The vector \(5v_1 = \begin{bmatrix} 100 \\ 2750 \end{bmatrix}\) represents the output of mine 1 after operating for 5 days, producing 100 metric tons of copper and 2750 kilograms of silver.

#### Part (b):
We want to find the number of days \(x_1\) and \(x_2\) that each mine operates to produce 150 metric tons of copper and 2825 kilograms of silver. This leads to the system of equations:
\[
20x_1 + 30x_2 = 150 \quad \text{(copper)}
\]
\[
550x_1 + 500x_2 = 2825 \quad \text{(silver)}
\]
We can solve this system using Gaussian elimination.

### Problem 5:
We are given the linear system:
\[
\begin{aligned}
x + 2y &= 7 \\
4x + hy &= k
\end{aligned}
\]
We need to determine the values of \(h\) and \(k\) that give (a) no solution, (b) a unique solution, and (c) infinitely many solutions.

#### (a) No Solution:
For the system to have no solution, the lines must be parallel but not coincident. This happens when the ratios of the coefficients of \(x\) and \(y\) are equal but the constants differ. Setting up the condition:
\[
\frac{1}{4} = \frac{2}{h} \quad \text{but} \quad \frac{7}{k} \neq \frac{1}{4}
\]
This leads to \(h = 8\), and for no solution, \(k \neq 14\).

#### (b) Unique Solution:
For a unique solution, the lines must intersect, which happens when the determinant of the coefficient matrix is nonzero:
\[
\text{Determinant: } 1 \cdot h - 2 \cdot 4 = h - 8 \neq 0
\]
Thus, for a unique solution, \(h \neq 8\).

#### (c) Infinitely Many Solutions:
For infinitely many solutions, the lines must coincide, which happens when:
\[
\frac{1}{4} = \frac{2}{h} = \frac{7}{k}
\]
This leads to \(h = 8\) and \(k = 14\).

### Problem 6:
Given vectors \(u = \begin{bmatrix} 2 \\ 1 \end{bmatrix}\) and \(v = \begin{bmatrix} 6 \\ 3 \end{bmatrix}\):
#### (a) Linear Combination:
The linear combination \(cu + dv\) is:
\[
cu + dv = c \begin{bmatrix} 2 \\ 1 \end{bmatrix} + d \begin{bmatrix} 6 \\ 3 \end{bmatrix} = \begin{bmatrix} 2c + 6d \\ c + 3d \end{bmatrix}
\]

Would you like more detailed steps for any of these problems?

Here are 5 related questions:
1. How do you represent a general solution using Gaussian elimination?
2. What does it mean geometrically if a system of equations has no solution?
3. How can Gaussian elimination help find the rank of a matrix?
4. How do you determine if a set of vectors spans \(\mathbb{R}^n\)?
5. What is the importance of the determinant in solving linear systems?

**Tip**: To avoid rounding errors in large systems, use fractions instead of decimals during elimination!

do questions 3 4 5 and 6 make sure to only use matrices and use gaussian elimination

Solve a set of linear algebra problems using Gaussian elimination, including questions on linear dependence, mine production, and linear systems. Detailed steps and solutions are provided for students in college-level math courses.

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