Linear Algebra Exam Practice Questions | Generated by AI
I understand you’re looking for typical past “big questions” (formalization and calculation problems) from the Guangdong Self-Study Exam Linear Algebra (Course Code: 02198) to use as practice.
Since the official English syllabus and past papers are hard to find, I’ll provide you with common types of big questions from a standard undergraduate linear algebra course, which are highly likely to cover the final, most involved questions on the 02198 exam. These questions usually test a deeper understanding and involve multiple steps.
Here are three major categories of big questions, along with specific exercise examples.
1. ⚙️ Calculation-Intensive Questions
These questions often involve large matrices and require a sequence of operations.
Exercise 1: Finding an Inverse Matrix and Solving a System
Given the matrix \(A = \begin{pmatrix} 1 & -1 & 0 \ 0 & 2 & 1 \ 1 & 0 & 3 \end{pmatrix}\).
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Calculate the inverse matrix \(A^{-1}\) using the adjugate matrix formula or the row reduction method \(\left(A I\right) \to \left(I A^{-1}\right)\). - Use the inverse matrix \(A^{-1}\) to solve the system of linear equations \(A\mathbf{x} = \mathbf{b}\), where \(\mathbf{b} = \begin{pmatrix} 2 \ 1 \ 5 \end{pmatrix}\).
Exercise 2: Change of Basis and Coordinate Transformation
In \(\mathbb{R}^3\), consider two bases:
- The standard basis \(E = {\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3}\).
- A new basis \(B = {\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3}\), where \(\mathbf{v}_1 = \begin{pmatrix} 1 \ 0 \ 1 \end{pmatrix}\), \(\mathbf{v}_2 = \begin{pmatrix} 1 \ 1 \ 0 \end{pmatrix}\), and \(\mathbf{v}_3 = \begin{pmatrix} 0 \ 1 \ 1 \end{pmatrix}\).
- Find the transition matrix \(P_{E \leftarrow B}\) from basis \(B\) to basis \(E\).
- Find the transition matrix \(P_{B \leftarrow E}\) from basis \(E\) to basis \(B\).
- If a vector \(\mathbf{w}\) has coordinates \([\mathbf{w}]_B = \begin{pmatrix} 2 \ -1 \ 3 \end{pmatrix}\) in basis \(B\), find its coordinates \([\mathbf{w}]_E\) in the standard basis \(E\).
2. 🔢 Eigenvalues and Diagonalization Questions
These are almost always the most important and multi-step questions.
Exercise 3: Diagonalization and Matrix Powers
Given the matrix \(A = \begin{pmatrix} 2 & 2 & -2 \ 2 & 2 & -2 \ -2 & -2 & 2 \end{pmatrix}\).
- Find all the eigenvalues \(\lambda\) of \(A\).
- For each eigenvalue, find the corresponding eigenspace and a set of linearly independent eigenvectors.
- Determine if \(A\) is diagonalizable. If it is, find the invertible matrix \(P\) and the diagonal matrix \(\Lambda\) such that \(P^{-1}AP = \Lambda\).
- Use the result to calculate the matrix power \(A^{100}\).
3. ✍️ Formalization/Proof and Conceptual Questions
These questions often involve properties of matrices or linear maps and require a rigorous argument, sometimes using definitions or theorems.
Exercise 4: Rank and Linear Systems (Formalization)
| Consider a system of \(m\) linear equations in \(n\) unknowns, \(A\mathbf{x} = \mathbf{b}\). Let \(A\) be the coefficient matrix and \(\tilde{A} = (A | \mathbf{b})\) be the augmented matrix. |
Formalization/Proof Question:
Prove that the system \(A\mathbf{x} = \mathbf{b}\) has:
- No solution if and only if \(Rank(A) < Rank(\tilde{A})\).
- A unique solution if and only if \(Rank(A) = Rank(\tilde{A}) = n\).
- Infinitely many solutions if and only if \(Rank(A) = Rank(\tilde{A}) < n\).
Exercise 5: Properties of Linear Maps (Formalization)
Let \(V\) and \(W\) be two finite-dimensional vector spaces, and let \(T: V \to W\) be a linear transformation.
Formalization/Proof Question:
Prove that if \(T\) is injective (one-to-one), then the nullity of \(T\) (the dimension of the kernel \(\text{Null}(T)\)) is zero. That is, \(\dim(\text{Null}(T)) = 0\).
Summary of Key Concepts Tested
The “big questions” typically involve a combination of these core topics:
- Matrix Operations: Determinants, Inverses, Adjugates.
- Linear Systems: Rank, Rouché–Capelli theorem, solving non-homogeneous systems.
- Vector Spaces: Basis, Dimension, Subspaces, Change of Basis.
- Eigen-Theory: Eigenvalues, Eigenvectors, Eigenspaces, Diagonalization, Quadratic Forms.
Would you like me to provide step-by-step solutions for any of these exercises, or should I try to find a few more examples focusing on Quadratic Forms?