Multiple Choice Question 2.Let U be any n-dimensional real inner-product space w

Question

Multiple Choice Question

2.Let U be any n-dimensional real inner-product space with arbitrary basis B = {u 1, u 2, ..., u n}

a.For u e U, then u = <u, u 1>u 1 + <u, u 2>u 2 + ... + <u, u n>u n.
b.If u = a1 u 1 + a2 u 2 + ... + an u n, and v = b1 u 1 + b2 u 2 + ... + bn u n, then <u,v> = a1b1 + a2b2 + ... + anbn
c.With u and v given in choice "b" above, <u, v> = 0, if and only if, u or v = 0.
d.If B is an orthonormal basis, choices "a", "b", and "c" are all true.
e.If B is an orthonormal basis, choices "a" and "b" are both true, but "c" is false.

Explanation / Answer

if b is an orthonormal basis, then (a) is true.
(b) is not necessarily true while the given inner product may be the standard inner product or any other inner product.
for instance, <a,b> =a1b1 + 2a2b2 is also an inner product on R2 but this is not the standard inner product. if B is an orthonormal basis, then every non zero pair of vectors are also orthogonal. that means <u, v> = 0 is not necessarily imply either u or v is the zero vector. so, if you are referring only standard inner product, then only the choice 'e' is true. other wise, only (a) is true. the remaining are depending on the conditions prevail. for any doubt, please send a message and i am ready to clear any doubt regarding this. thank you.

Related Questions

Navigate

• Browse (All)
• Browse M
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.