ENGINEERING MATHEMATICS – UNIT 4:
APPLICATIONS OF REAL INNER PRODUCT SPACE
Time: 3 Hours Max Marks: 100
Part A – (2 Marks each)
Answer all questions (10 × 2 = 20 marks)
1. Define a real inner product space.
2. Write the formula for the projection of a vector u on v.
3. What is meant by least squares approximation?
4. Define orthogonal projection.
5. Write the normal equation for least squares solution of A x = b.
6. What is an orthogonal matrix?
7. Define orthonormal basis in R■.
8. What is the geometrical meaning of projection?
9. Define the concept of best approximation in inner product space.
10. State the Pythagoras theorem for an inner product space.
Part B – (5 Marks each)
Answer any FIVE questions (5 × 5 = 25 marks)
1. Find the projection of u = (3,4) onto v = (1,2).
2. Find the least squares solution of A x = b, where A = [[1,1],[1,-1],[1,1]], b = [2,0,2].
3. State and prove the orthogonal decomposition theorem.
4. Determine an orthonormal basis for the subspace spanned by (1,1,0) and (1,0,1).
5. Show that the least squares solution minimizes ||A x - b||.
6. Find the projection of b = (3,1,2) onto the subspace spanned by (1,0,1) and (0,1,1).
7. Verify that for any two orthogonal vectors u and v, ||u+v||² = ||u||² + ||v||².
Part C – (11 Marks each)
Answer any THREE questions (3 × 11 = 33 marks)
1. (a) Explain the concept of best approximation in a real inner product space.
(b) Prove that the projection of a vector b onto a subspace W gives the best approximation to b in
W.
2. Find the least squares solution of the overdetermined system x + y = 2, 2x + y = 3, 3x + y = 4.
3. Apply Gram-Schmidt orthogonalization to find an orthonormal basis for the subspace spanned by
(1,1,1), (1,1,0), (1,0,0).
4. Prove that if {u■, u■, …, u■} is an orthonormal set in a real inner product space, then for any
vector x, ||x||² = Σ |■x, u■■|² (Parseval’s Identity).
5. (a) Define normal equations for a least squares problem.
(b) Use them to find the best fit line y = ax + b for the data points (1,2), (2,3), (3,5).
Part D – (21 Marks)
APPLICATIONS OF REAL INNER PRODUCT SPACE
Time: 3 Hours Max Marks: 100
Part A – (2 Marks each)
Answer all questions (10 × 2 = 20 marks)
1. Define a real inner product space.
2. Write the formula for the projection of a vector u on v.
3. What is meant by least squares approximation?
4. Define orthogonal projection.
5. Write the normal equation for least squares solution of A x = b.
6. What is an orthogonal matrix?
7. Define orthonormal basis in R■.
8. What is the geometrical meaning of projection?
9. Define the concept of best approximation in inner product space.
10. State the Pythagoras theorem for an inner product space.
Part B – (5 Marks each)
Answer any FIVE questions (5 × 5 = 25 marks)
1. Find the projection of u = (3,4) onto v = (1,2).
2. Find the least squares solution of A x = b, where A = [[1,1],[1,-1],[1,1]], b = [2,0,2].
3. State and prove the orthogonal decomposition theorem.
4. Determine an orthonormal basis for the subspace spanned by (1,1,0) and (1,0,1).
5. Show that the least squares solution minimizes ||A x - b||.
6. Find the projection of b = (3,1,2) onto the subspace spanned by (1,0,1) and (0,1,1).
7. Verify that for any two orthogonal vectors u and v, ||u+v||² = ||u||² + ||v||².
Part C – (11 Marks each)
Answer any THREE questions (3 × 11 = 33 marks)
1. (a) Explain the concept of best approximation in a real inner product space.
(b) Prove that the projection of a vector b onto a subspace W gives the best approximation to b in
W.
2. Find the least squares solution of the overdetermined system x + y = 2, 2x + y = 3, 3x + y = 4.
3. Apply Gram-Schmidt orthogonalization to find an orthonormal basis for the subspace spanned by
(1,1,1), (1,1,0), (1,0,0).
4. Prove that if {u■, u■, …, u■} is an orthonormal set in a real inner product space, then for any
vector x, ||x||² = Σ |■x, u■■|² (Parseval’s Identity).
5. (a) Define normal equations for a least squares problem.
(b) Use them to find the best fit line y = ax + b for the data points (1,2), (2,3), (3,5).
Part D – (21 Marks)
