A First Course in Linear Algebra, 2017a

4.11. Orthogonality and the Gram Schmidt Process 255
Exercise 4.11.13 Using the Gram Schmidt process find an orthonormal basis for the following span:
⎧⎡
⎤ ⎡ ⎤ ⎡ ⎤⎫
⎨ 1 2 1 ⎬
span ⎣ 2 ⎦, ⎣ −1 ⎦, ⎣ 0 ⎦
⎩
⎭
1 3 0
Exercise 4.11.14 Using the Gram Schmidt process find an orthonormal basis for the following span:
⎧⎡
⎤ ⎡ ⎤ ⎡ ⎤⎫
1 2 1
⎪⎨
span ⎢ 2
⎥
⎣ 1 ⎦
⎪⎩
, ⎢ −1
⎥
⎣ 3 ⎦ , ⎢ 0
⎪⎬
⎥
⎣ 0 ⎦
⎪⎭
0 1 1
⎧⎡
⎨
Exercise 4.11.15 The set V = ⎣
⎩
for this subspace.
Exercise 4.11.16 Consider the following scalar equation of a plane.
x
y
z
⎤
⎫
⎬
⎦ :2x + 3y − z = 0
⎭ is a subspace of R3 . Find an orthonormal basis
2x − 3y + z = 0
⎡ ⎤
3
Find the orthogonal complement of the vector⃗v = ⎣ 4 ⎦. Also find the point on the plane which is closest
1
to (3,4,1).
Exercise 4.11.17 Consider the following scalar equation of a plane.
x + 3y + z = 0
⎡ ⎤
1
Find the orthogonal complement of the vector⃗v = ⎣ 2 ⎦. Also find the point on the plane which is closest
1
to (3,4,1).
Exercise 4.11.18 Let ⃗v be a vector and let ⃗n be a normal vector for a plane through the origin. Find the
equation of the line through the point determined by⃗v which has direction vector⃗n. Show that it intersects
the plane at the point determined by⃗v −proj ⃗n ⃗v. Hint: The line:⃗v+t⃗n. It is in the plane if⃗n•(⃗v +t⃗n)=0.
Determine t. Then substitute in to the equation of the line.
Exercise 4.11.19 As shown in the above problem, one can find the closest point to⃗v in a plane through the
origin by finding the intersection of the line through⃗v having direction vector equal to the normal vector
to the plane with the plane. If the plane does not pass through the origin, this will still work to find the
point on the plane closest to the point determined by⃗v. Here is a relation which defines a plane
2x + y + z = 11