1. Define a vector space and provide examples.
2. Prove that the set of all real-valued functions forms a vector space.
3. What is a subspace? Give examples and non-examples.
4. Prove that the intersection of two subspaces is a subspace.
5. Define linear independence and dependence of vectors. Provide examples.
6. Find a basis for the vector space \( \mathbb{R}^3 \).
7. State and prove the Rank-Nullity thereom
### **Unit 2: Linear Transformations**
1. Define a linear transformation and provide examples.
2. Prove that the kernel and image of a linear transformation are subspaces.
3. Find the matrix representation of a given linear transformation.
4. Determine whether a given transformation is injective, surjective, or bijective.
5. State and prove the Dimension Theorem for linear transformations.
6. Compute the rank and nullity of a given matrix.
**Unit 3: Matrices and Determinants**
1. Define the determinant of a matrix and explain its properties.
2. Compute the determinant of a 3x3 or 4x4 matrix.
3. Prove that the determinant of a product of matrices is the product of their determinants.
4. Define the adjoint and inverse of a matrix. Find the inverse of a given matrix.
5. Solve a system of linear equations using Cramer's Rule.
6. Define eigenvalues and eigenvectors. Find the eigenvalues and eigenvectors of a given matrix.
Unit 4: Inner Product Spaces**
1. Define an inner product space and provide examples.
2. Prove the Cauchy-Schwarz inequality in an inner product space.
3. Define orthogonality and orthogonal projection. Find the orthogonal projection of a vector onto a subspace.
4. Apply the Gram-Schmidt process to orthogonalize a set of vectors.
5. Define a symmetric matrix and prove that its eigenvalues are real.
6. Diagonalize a given symmetric m
atrix.
### **Unit 5: Canonical Forms**
1. Define the Jordan canonical form of a matrix.
2. Find the Jordan canonical form of a given matrix.
3. Define the minimal polynomial of a matrix and explain its significance.
4. Prove that every square matrix satisfies its characteristic equation (Cayley-Hamilton Theorem).
5. Define the rational canonical form of a matrix.
# **Practice Problems**
1. Solve systems of linear equations using matrix methods.
2. Find the eigenvalues and eigenvectors of a given matrix.
3. Diagonalize a matrix, if possible.
4. Compute the determinant, rank, and inverse of a matrix.
5. Apply the Gram-Schmidt process to a set of vectors.
సమాధానాల కోసం ఇక్కడ క్లిక్ చెయ్యండి.
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