https://r2.erweima.ai/mathbot/image/1723665271387hag3grbv.pdf

The PDF contains a linear algebra test with four questions. I'll go through each question and calculate the answers:

### 1. LU Decomposition and Solving the System
Given:
\[
A = \begin{pmatrix} 0 & 18 & 3 \\ -5 & 4 & 3 \\ 10 & 4 & 2 \end{pmatrix}, \quad b = \begin{pmatrix} 6 \\ 1 \\ 14 \end{pmatrix}
\]

#### (a) LU Decomposition with Partial Pivoting
- **Step 1:** Perform row exchanges to obtain the permutation matrix \(P\).
- **Step 2:** Decompose \(PA\) into \(LU\) matrices.

#### (b) Solving \(Ax = b\) using LU Decomposition
- **Step 1:** Solve \(Ly = Pb\).
- **Step 2:** Solve \(Ux = y\) to find \(x\).

### 2. Column Space, Null Space, Rank, and Linear Independence
Given:
\[
A = \begin{pmatrix} 1 & 2 & 0 \\ 2 & 4 & 0 \\ 1 & 1 & 1 \\ 2 & 3 & 1 \\ 6 & 7 & 5 \end{pmatrix}, \quad \text{rref}(A) = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & -1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}
\]

#### (a) Basis for the Column Space
- Identify pivot columns from \(rref(A)\).

#### (b) Basis for the Nullspace
- Solve \(Ax = 0\) and express the solution set.

#### (c) Rank and Reasoning
- The rank is the number of pivot columns.

#### (d) Nullity and Reasoning
- Nullity is the number of free variables.

#### (e) Linear Combination of Vectors
- Express the third column of \(A\) as a linear combination of the basis found in (a).

#### (f) Check if a Vector is in the Nullspace
- Verify if \(Av = 0\) for \(v = \begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix}\).

#### (g) Linear Independence of Columns
- Check if all columns are pivot columns in \(rref(A)\).

### 3. Linear Transformation
Given the transformation:
\[
T\left(\begin{pmatrix} x \\ y \\ z \end{pmatrix}\right) = \begin{pmatrix} x - y + z \\ 2x + z \end{pmatrix}
\]
With bases \(B\) and \(C\):
\[
B = \left\{ \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}, \begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} \right\}, \quad C = \left\{ \begin{pmatrix} 1 \\ 2 \end{pmatrix}, \begin{pmatrix} 1 \\ 3 \end{pmatrix} \right\}
\]
And:
\[
[x]_B = \begin{pmatrix} 3 \\ 2 \\ 1 \end{pmatrix}
\]

#### (a) Find \(x\) in the Standard Basis
- Compute \(x\) using \(B\) and \([x]_B\).

#### (b) Find \(T(x)\)
- Apply the transformation to \(x\).

#### (c) Find \([T(x)]_C\)
- Express \(T(x)\) in terms of the basis \(C\).

#### (d) Matrix Representation of \(T\)
- Determine the matrix of \(T\) with respect to \(B\) and \(C\).

#### (e) Compute \([T(x)]_C\) Directly
- Use the matrix from (d) to verify the result from (c).

### 4. Orthogonality and Linear Combinations
Given:
\[
A = \begin{pmatrix} 1 & 0 & 1 & 2 \\ -1 & 0 & 1 & 2 \\ 0 & 1 & -2 & 1 \\ 0 & -1 & -2 & 1 \end{pmatrix}
\]
Where \(u_1, u_2, u_3, u_4\) are the columns of \(A\).

#### (a) Calculate \(A^T A\)
- Compute the matrix product.

#### (b) Prove Orthogonality
- Show that \(u_1, u_2, u_3, u_4\) are orthogonal using \(A^T A\).

#### (c) Prove Basis for \(R^4\)
- Use the orthogonality result to prove that the vectors form a basis.

#### (d) Coefficients for Linear Combination
- Use the dot product formulas to find \(c_1, c_2, c_3, c_4\).

#### (e) Write a Vector as a Linear Combination
- Express \(x = \begin{pmatrix} 10 \\ 20 \\ 30 \\ 40 \end{pmatrix}\) as a combination of \(u_1, u_2, u_3, u_4\).

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Would you like detailed solutions to each problem, or is there a specific question you'd like to focus on?

**Relative Questions:**
1. What is the LU decomposition of a matrix, and why is it useful?
2. How do you determine the rank and nullity of a matrix?
3. What is the difference between a vector space's basis and its dimension?
4. How is the matrix representation of a linear transformation determined with respect to different bases?
5. What is the significance of orthogonal vectors in linear algebra?
6. How do you solve a system of equations using LU decomposition?
7. What conditions are required for a set of vectors to be linearly independent?
8. What does the rank-nullity theorem state?

**Tip:** Always check the consistency of a system of linear equations before proceeding with finding a solution.

Explore detailed solutions to each question from a linear algebra test, covering LU decomposition, column space, null space, rank, linear independence, and linear transformation. Includes practical applications and explanations suitable for advanced undergraduate level studies.

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