Orthogonal basis

Question 1

the orthogonal complement of a line through the origin

Answer 1

is the plane through the origin perpendicular to it

Question 2

f two planes being perpendicular does not correspond to the orthogonal complement,

Answer 2

since in three dimensions a pair of vectors, one from each of a pair of perpendicular planes, might meet at any angle.

Question 3

In four-dimensional Euclidean space, the orthogonal complement of a line

Answer 3

a hyperplane and vice versa, and that of a plane is a plane

Question 4

The orthogonal complement of a subspace

Answer 4

is the space of all vectors that are orthogonal to every vector in the subspace

Question 5

An orthogonal matrix

Answer 5

is a matrix whose column vectors are orthonormal to each other.

Question 6

Significance of orthogonal vectors basis

Answer 6

If there is a basis, we don’t want to search for a c1,c2,c3

Question 7

orthogonal set

Orthonormal set

Answer 7

set of all different vectors that are perpendicular to each other

orthogonal set of unit vectors

Question 8

Making of orthonormal vectors if it is orthogonal

Answer 8

replacing each vector unit vector in the direction of each vector

Question 9

Orthogonal sets are___.

Answer 9

linearly independent

Question 10

linearly indep proof - c1u1 + c2u2 + .. = 0

what are the 2 steps?

Answer 10

1. multiply both sides by u1
2. since u1,ui i>1 cmes u1.ui !=0
   so c1 = c2 = …=0

Question 11

projection formula

Answer 11

to project vector in Rn to a
subspace given its
orthogonal basis

Question 12

if basis is orthonormal, then ui.ui , projction formula

Answer 12

xw = (x.u1)u1 +(x.u2).u2 + ..

Question 13

Problems:
Given ortho basis B or span of W, given a vector
find projection of the vector to that subspace

Answer 13

DONT PANIC!!YOU WILL DO IT RIGHT

Question 14

B coordinates?

Answer 14

[x]w = (x.u1/u1.u1) , (x.u2/u2.u2) , …

Question 15

cosine of angle

Answer 15

cos theta = v.u/norm u . normv