Linear Algebra

Section VI. Projection 267
(d) Find the orthogonal projection of �v into P by keeping only the P part from
the prior item.
(e) Check that against the result from applying Theorem 3.8.
� 3.13 We have three ways to find the orthogonal projection of a vector into a line,
the Definition 1.1 way from the first subsection of this section, the Example 3.2
and 3.3 way of representing the vector with respect to a basis for the space and
then keeping the M part, and the way of Theorem 3.8. For these cases, do all
three ways. � � � �
1
x ��
(a) �v = , M = { x + y =0}
−3
y
� � � �
0
x ��
(b) �v = 1 , M = { y x + z = 0 and y =0}
2
z
3.14 Check that the operation of Definition 3.1 is well-defined. That is, in Example
3.2 and 3.3, doesn’t the answer depend on the choice of bases?
3.15 What is the orthogonal projection into the trivial subspace?
3.16 What is the projection of �v into M along N if �v ∈ M?
3.17 Show that if M ⊆ R n is a subspace with orthonormal basis 〈�κ1,... ,�κn〉 then
the orthogonal projection of �v into M is this.
(�v �κ1) · �κ1 + ···+(�v �κn) · �κn
� 3.18 Prove that the map p: V → V is the projection into M along N if and only
if the map id − p is the projection into N along M. (Recall the definition of the
difference of two maps: (id − p)(�v) =id(�v) − p(�v) =�v − p(�v).)
� 3.19 Show that if a vector is perpendicular to every vector in a set then it is
perpendicular to every vector in the span of that set.
3.20 True or false: the intersection of a subspace and its orthogonal complement is
trivial.
3.21 Show that the dimensions of orthogonal complements add to the dimension
oftheentirespace.
� 3.22 Suppose that �v1,�v2 ∈ R n are such that for all complements M,N ⊆ R n ,the
projections of �v1 and �v2 into M along N are equal. Must �v1 equal �v2? (If so, what
if we relax the condition to: all orthogonal projections of the two are equal?)
� 3.23 Let M,N be subspaces of R n . The perp operator acts on subspaces; we can
ask how it interacts with other such operations.
(a) Show that two perps cancel: (M ⊥ ) ⊥ = M.
(b) Prove that M ⊆ N implies that N ⊥ ⊆ M ⊥ .
(c) Show that (M + N) ⊥ = M ⊥ ∩ N ⊥ .
� 3.24 The material in this subsection allows us to express a geometric relationship
that we have not yet seen between the rangespace and the nullspace of a linear
map.
(a) Represent f : R 3 → R given by
� �
v1
v2
v3
↦→ 1v1 +2v2 +3v3