In all parts of this problem, let V be the subspace of allvectors x → in ℝ 4 such that x 3 = x 1 + x 2 and x 4 = x 2 + x 3 .See Problems 72 and 73 of Section 4.3. a. Find the matrix P V of the orthogonal projection ontothe subspace V in ℝ 4 . Hint: Work with one of thebases of V we considered in Problem 4.3.73. b. What is the relationship between the subspaces Wand V defined in Exercises 69 and 70? Consequently, what is the relationship between the matrices P W and P V in Exercises 69 and 70?
In all parts of this problem, let V be the subspace of allvectors x → in ℝ 4 such that x 3 = x 1 + x 2 and x 4 = x 2 + x 3 .See Problems 72 and 73 of Section 4.3. a. Find the matrix P V of the orthogonal projection ontothe subspace V in ℝ 4 . Hint: Work with one of thebases of V we considered in Problem 4.3.73. b. What is the relationship between the subspaces Wand V defined in Exercises 69 and 70? Consequently, what is the relationship between the matrices P W and P V in Exercises 69 and 70?
Solution Summary: The author explains how to find the matrix P_v of the orthogonal projection onto V.
In all parts of this problem, let V be the subspace of allvectors
x
→
in
ℝ
4
such that
x
3
=
x
1
+
x
2
and
x
4
=
x
2
+
x
3
.See Problems 72 and 73 of Section 4.3. a. Find the matrix
P
V
of the orthogonal projection ontothe subspace V in
ℝ
4
. Hint: Work with one of thebases of V we considered in Problem 4.3.73. b. What is the relationship between the subspaces Wand V defined in Exercises 69 and 70? Consequently, what is the relationship between the matrices
P
W
and
P
V
in Exercises 69 and 70?
I need help for problem (h). Check that the set at (h) is a subspace of Rn or not.
Do questions 53 and 54
Show if it is a subspace using these 3 steps:
1. has to be equal to the 0 vector
2. has to be closed under addition
3. has to be closed under mulitplication
Suppose that S1 and S2 are subspaces of a vector space (V, F). Show that their intersection S1 ∩ S2 is also a subspace of (V, F). Is their union S1 ∪ S2 always a subspace?
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