MAT211 Lecture 14
Orthogonal projections and orthogonal basis
‣
Orthogonality, length, unit vectors
‣
Orthonormal vectors: defnition and properties
‣
Orthogonal projections: defnition, Formula and properties.
‣
Orthogonal complements
‣
Pythagorean theorem, Cauchy inequality, angle between
two vectors
Defnition
Two vectors u and v in R
n
are
perpendicular or
orthogonal
iF u.v=0
The
length
oF a vector v in R
n
is v=
!
v.v
A vector v in R
n
is called a
unit vector
iF
v=1
Example
±ind a unit vector in the line oF
multiples oF (1,1,3)
±ind a vector oF length 2 orthogonal
to (1,1,3)
Defnition
A vector v in R
n
is orthogonal to a subspace
V oF R
n
iF it is orthogonal to all vectors in V
Remark: IF (b
1
,b
2
,..,b
m
) is a
basis oF V, then v is orthogonal
to V iF (and only iF) v is
orthogonal to b
1
,b
2
,.. and b
m .
EXAMPLE
Consider the subspace V oF R
3
span by
(1,1,1) and (1,0,1).
±ind all the vectors orthogonal to V.
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 Fall '08
 MOVSHEV
 Linear Algebra, Pythagorean Theorem, Vectors, ‣Pythagorean theorem

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