https://file.aiquickdraw.com/mathbot/file/17266635136858gjqilnz.pdf

Jag vill att du ska lösa:
Uppgift A (5)
Uppgift B (3)
Uppgift C (3)
Uppgift D (6)

Here is a breakdown of the tasks you requested:

### **Uppgift A (5)**
This task involves solving a system of linear equations:

\[
\begin{cases}
2x + y - 4z = -2 \\
x - 3y - 9z = 6 \\
x + ay + bz = -2
\end{cases}
\]

To solve this, follow these steps:
1. Express the system in matrix form.
2. Perform Gaussian elimination to reduce the system.
3. Determine the values of \(a\) and \(b\) for which the system has infinitely many solutions (this occurs when the system is underdetermined).
4. Solve for the general solution once \(a\) and \(b\) are known.

### **Uppgift B (3)**
This is a basis change problem, where we need to express the new coordinates \((x'_1, x'_2, x'_3)\) in terms of the original coordinates \((x_1, x_2, x_3)\), given the basis transformation:

\[
\begin{cases}
e'_1 = e_1 + 3e_2 + 3e_3 \\
e'_2 = 2e_1 + e_2 \\
e'_3 = e_2 + e_3
\end{cases}
\]

We must:
1. Set up the transformation matrix from the old basis to the new one.
2. Invert the transformation matrix to express the new coordinates in terms of the old ones.

### **Uppgift C (3)**
This problem involves finding the point where a normal line through a given point \(P\) intersects the plane:

Plane equation: \(2x + y - 2z + 1 = 0\)

Given \(P = (-3, 0, 2)\), the steps are:
1. Find the normal vector to the plane.
2. Write the parametric equation of the line passing through \(P\) in the direction of the normal.
3. Solve for the parameter where the line intersects the plane.
4. Compute the coordinates of the intersection.

### **Uppgift D (6)**
Here, you are asked to calculate the area of a triangle with vertices:
- One at \((1, 2, 3)\),
- Another at the mirror image of \(P = (13, 10, 7)\) across the plane \(x - z = 0\),
- The third at the point where the plane \(\pi: x + 2y - 2z = -4\) intersects the line given by:

\[
\begin{cases}
x = 6 \\
y = t \\
z = 17 - 2t
\end{cases}
\]

The steps to solve this are:
1. Find the mirror image of \(P\) across the plane.
2. Find the intersection of the line with the plane \(\pi\).
3. Use the three vertices to compute the area of the triangle using the cross-product method.

Let me know if you'd like me to solve any specific step in detail or go through the solution process further!

### Relative questions:
1. How can we determine if a system of equations has no solutions or infinitely many?
2. What is the importance of the transformation matrix in basis change problems?
3. How do you derive the equation of a line from a point and a normal vector?
4. How do you compute the mirror image of a point across a plane in 3D space?
5. What is the role of the cross-product in finding the area of a triangle?

#### Tip:
When working with systems of equations, always check for special cases such as parallel planes or dependent equations, which may indicate infinitely many solutions or no solutions at all.

Jag vill att du ska lösa: Uppgift A (5) Uppgift B (3) Uppgift C (3) Uppgift D (6)

Explore detailed solutions to linear systems with Gaussian elimination, basis change problems, and the calculation of triangle areas using normal vectors and cross-product methods. Suitable for university students studying linear algebra.

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