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5. (a) Let V = R* and let (·, ·) be the standard dot product on V. (i) Calculate (e, +€3; €2 + €4). 12 (ii) If u = EV, show that (u, u) > 0. (iii) Give an example of a vector of length 2. (iv) Let u = e1 +€3. Find S = {v € V : (u, v) = 0}, (b) If V = R", and U is a subspace of V prove that U- is a subspace of V, where U- = {v € V : (u, v) = 0 for all u E U} (c) If V = R" and U is a subspace of V with dim(U) = m, state the dimension of U+. (d) Use the Gram-Schmidt process to find an orthonormal basis of R3 that contains a scalar multiple of the vector v1

5. (a) Let V = R* and let (·, ·) be the standard dot product on V. (i) Calculate (e, +€3; €2 + €4). 12 (ii) If u = EV, show that (u, u) > 0. (iii) Give an example of a vector of length 2. (iv) Let u = e1 +€3. Find S = {v € V : (u, v) = 0}, (b) If V = R", and U is a subspace of V prove that U- is a subspace of V, where U- = {v € V : (u, v) = 0 for all u E U} (c) If V = R" and U is a subspace of V with dim(U) = m, state the dimension of U+. (d) Use the Gram-Schmidt process to find an orthonormal basis of R3 that contains a scalar multiple of the vector v1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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5. (a) Let V = R* and let (•, ·) be the standard dot product on V. (i) Calculate (e + e3, €2 + €4). (ii) If u = E V, show that (u, u) > 0. x3 (iii) Give an example of a vector of length 2. (iv) Let u = e1 +e3. Find S = {v € V : (u, v) = 0}, (b) If V = R", and U is a subspace of V prove that U- is a subspace of V, where U- = {v € V : (u, v) = 0 for all u E U} (c) If V = R" and U is a subspace of V with dim(U) = m, state the dimension of U+. (d) Use the Gram-Schmidt process to find an orthonormal basis of R3 that contains a scalar multiple of the vector v1 =
Transcribed Image Text:5. (a) Let V = R* and let (•, ·) be the standard dot product on V. (i) Calculate (e + e3, €2 + €4). (ii) If u = E V, show that (u, u) > 0. x3 (iii) Give an example of a vector of length 2. (iv) Let u = e1 +e3. Find S = {v € V : (u, v) = 0}, (b) If V = R", and U is a subspace of V prove that U- is a subspace of V, where U- = {v € V : (u, v) = 0 for all u E U} (c) If V = R" and U is a subspace of V with dim(U) = m, state the dimension of U+. (d) Use the Gram-Schmidt process to find an orthonormal basis of R3 that contains a scalar multiple of the vector v1 =
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