Exercise 2. 1) Suppose T: V →V is an operator on the inner product space V, with dim V = 4. Suppose dim(null T) = 1, what is the dimension of (range T)+? 2) Consider R? with its Euclidean inner product. Suppose and v If U = Span(v) and Po(u) = v, what does b have to be equal to? Answer: 2 3) Consider R' with its Euclidean inner product. Let L be the line spanned by the () - (3) vector 2 in R³. If & = -1), which one of the following vectors is the orthogonal projection Pv of i on L? a) (2 b) (4 c) 3 d) 30 20

Exercise 2. 1) Suppose T: V →V is an operator on the inner product space V, with dim V = 4. Suppose dim(null T) = 1, what is the dimension of (range T)+? 2) Consider R? with its Euclidean inner product. Suppose and v If U = Span(v) and Po(u) = v, what does b have to be equal to? Answer: 2 3) Consider R' with its Euclidean inner product. Let L be the line spanned by the () - (3) vector 2 in R³. If & = -1), which one of the following vectors is the orthogonal projection Pv of i on L? a) (2 b) (4 c) 3 d) 30 20

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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Question

Exercise 2. 1) Suppose T : V → V is an operator on the inner product space V, with
dim V = 4. Suppose dim(null T) = 1, what is the dimension of (range T)+?
2) Consider R? with its Euclidean inner product. Suppose
u =
and v =
If U = Span(v) and Pu(u) = v, what does b have to be equal to?
Answer: 2
1
3) Consider R' with its Euclidean inner product. Let L be the line spanned by the
vector (2 in R³. If ū =
which one of the following vectors is the orthogonal
projection PLv of ī on L?
(10)
d) ( 20
30
a) (2
b) (4
c) 3
4) Suppose u, v E V, where V is an inner product space, and ||u|| = || || = 1 and (u, v) = 1.
Find u – v. 5) Suppose U and W are finite-dimensional subspaces of the inner product
space V, with WCU. Find Pw Py.

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