Exam prep

Important Topics:

1. Systems of Linear Equations

   • Solving using Gaussian elimination.
   • Understanding solution sets and parametric vector forms.

2. Matrix Operations

   • Matrix multiplication, inverses, and determinants.
   • Elementary row operations and row echelon form (REF).

3. Vector Spaces and Subspaces

   • Basis and dimension.
   • Null space, column space, and row space.

4. Eigenvalues and Eigenvectors

   • Finding eigenvalues and eigenvectors.
   • Applications in diagonalization and the characteristic polynomial.

5. Diagonalization

   • Conditions for diagonalizability.
   • Constructing the diagonal matrix and the matrix of eigenvectors.

6. Orthogonality and Projections

   • Orthogonal sets and orthonormal bases.
   • Projection onto a subspace.

7. Linear Transformations

   • Definitions and examples.
   • Kernel and range of a transformation.

Relevant Exam-Style Questions and Exercises:

1. Solving Systems of Linear Equations

   • Given a matrix $A$ and a vector $\mathbf{b}$, determine all $\mathbf{x}\in {\mathbb{R}}^4$ such that $T(\mathbf{x})=\mathbf{b}$, where $T(\mathbf{x})=A\mathbf{x}$.

2. Matrix Invertibility and Determinants

   • For a given matrix $A$, find values of $p$ for which $A$ is not invertible by calculating $\text{det}(A)$.

3. Eigenvalues and Eigenvectors

   • Determine the eigenvalues of a given matrix $A$. For each eigenvalue, find the corresponding eigenvectors.

4. Diagonalization

   • Verify if a given matrix $A$ is diagonalizable. If so, find a diagonal matrix $D$ and a matrix $P$ such that $A=PD{P}^{-1}$.

5. Orthogonality

   • Provide an example of a subset $H$ of ${\mathbb{R}}^2$ that satisfies certain properties related to orthogonality and vector operations.

6. Vector Spaces

   • Given matrices representing linear transformations, determine if they can be considered linear transformations based on provided conditions.