ENGINEERING MATHEMATICS – UNIT 3: INNER
PRODUCT SPACE
Time: 3 Hours Max Marks: 100
Part A – (2 Marks each)
Answer all questions (10 × 2 = 20 marks)
1. Define inner product on a vector space.
2. State the properties of inner product.
3. Define norm of a vector.
4. What is meant by orthogonal vectors?
5. Define orthonormal set.
6. What is the Cauchy-Schwarz inequality?
7. Define projection of a vector onto another vector.
8. Write the formula for distance between two vectors in an inner product space.
9. If u = (1,2) and v = (3,4), find <u,v>.
10. Define Gram-Schmidt orthogonalization process.
Part B – (5 Marks each)
Answer any FIVE questions (5 × 5 = 25 marks)
1. Verify that ■u,v■ = u■v■ + u■v■ + u■v■ defines an inner product on ■³.
2. Find the norm and distance between u = (1,2,3) and v = (2,1,2).
3. Show that (1,1,0) and (1,-1,0) are orthogonal vectors in ■³.
4. Apply the Gram-Schmidt process to the set {(1,1,0), (1,0,1)}.
5. State and prove the Pythagoras theorem in inner product spaces.
6. Find the projection of u = (2,1) onto v = (1,3).
7. Verify the Cauchy-Schwarz inequality for u=(2,1), v=(1,2).
Part C – (11 Marks each)
Answer any THREE questions (3 × 11 = 33 marks)
1. (a) Define inner product space and prove the parallelogram law.
(b) If u=(1,2,2) and v=(2,1,-1), find ■u,v■, ||u||, and ||v||.
2. Apply the Gram-Schmidt orthogonalization process to the vectors v■=(1,1,1), v■=(1,-1,0), and
v■=(1,0,-1).
3. (a) State and prove the Cauchy-Schwarz inequality.
(b) Deduce the Triangle inequality from it.
4. Let S = {(1,0,0), (1,1,0), (1,1,1)}. Apply Gram-Schmidt to obtain an orthonormal basis.
5. (a) Define orthogonal complement.
(b) If W = span{(1,1,0)} in ■³, find W⊥.
Part D – (21 Marks)
Answer ONE full question (1 × 21 = 21 marks)
PRODUCT SPACE
Time: 3 Hours Max Marks: 100
Part A – (2 Marks each)
Answer all questions (10 × 2 = 20 marks)
1. Define inner product on a vector space.
2. State the properties of inner product.
3. Define norm of a vector.
4. What is meant by orthogonal vectors?
5. Define orthonormal set.
6. What is the Cauchy-Schwarz inequality?
7. Define projection of a vector onto another vector.
8. Write the formula for distance between two vectors in an inner product space.
9. If u = (1,2) and v = (3,4), find <u,v>.
10. Define Gram-Schmidt orthogonalization process.
Part B – (5 Marks each)
Answer any FIVE questions (5 × 5 = 25 marks)
1. Verify that ■u,v■ = u■v■ + u■v■ + u■v■ defines an inner product on ■³.
2. Find the norm and distance between u = (1,2,3) and v = (2,1,2).
3. Show that (1,1,0) and (1,-1,0) are orthogonal vectors in ■³.
4. Apply the Gram-Schmidt process to the set {(1,1,0), (1,0,1)}.
5. State and prove the Pythagoras theorem in inner product spaces.
6. Find the projection of u = (2,1) onto v = (1,3).
7. Verify the Cauchy-Schwarz inequality for u=(2,1), v=(1,2).
Part C – (11 Marks each)
Answer any THREE questions (3 × 11 = 33 marks)
1. (a) Define inner product space and prove the parallelogram law.
(b) If u=(1,2,2) and v=(2,1,-1), find ■u,v■, ||u||, and ||v||.
2. Apply the Gram-Schmidt orthogonalization process to the vectors v■=(1,1,1), v■=(1,-1,0), and
v■=(1,0,-1).
3. (a) State and prove the Cauchy-Schwarz inequality.
(b) Deduce the Triangle inequality from it.
4. Let S = {(1,0,0), (1,1,0), (1,1,1)}. Apply Gram-Schmidt to obtain an orthonormal basis.
5. (a) Define orthogonal complement.
(b) If W = span{(1,1,0)} in ■³, find W⊥.
Part D – (21 Marks)
Answer ONE full question (1 × 21 = 21 marks)
