Welcome to the exercises section for Linear Algebra! Here, you'll find a variety of practice problems to reinforce your understanding of the subject.
Exercises
- Exercises on Matrices
- Exercises on Vector Spaces
- Exercises on Determinants
- Exercises on Eigenvalues and Eigenvectors
Matrices
- Problem 1: If ( A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} ), find ( A^2 ).
- Problem 2: Compute the inverse of ( A = \begin{pmatrix} 2 & 1 \ 1 & 2 \end{pmatrix} ).
More on Matrix Operations
Vector Spaces
- Problem 3: Determine whether the set of all ( 2 \times 2 ) matrices with real entries is a vector space.
- Problem 4: Prove that the set of all polynomials of degree less than or equal to 2 forms a vector space.
Deepen Your Understanding of Vector Spaces
Determinants
- Problem 5: Calculate the determinant of ( \begin{pmatrix} 1 & 2 & 3 \ 4 & 5 & 6 \ 7 & 8 & 9 \end{pmatrix} ).
- Problem 6: Use the determinant to find the area of a parallelogram spanned by vectors ( \vec{u} ) and ( \vec{v} ).
Explore Determinants in Depth
Eigenvalues and Eigenvectors
- Problem 7: Find the eigenvalues and eigenvectors of ( A = \begin{pmatrix} 4 & 1 \ 2 & 3 \end{pmatrix} ).
- Problem 8: Use eigenvalues to diagonalize ( A = \begin{pmatrix} 1 & 2 \ 3 & 4 \end{pmatrix} ).
Learn About Diagonalization and Eigenvalues
Linear Algebra Formula