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Orthogonal projections
- Intro Lesson: a14:35
- Intro Lesson: b3:50
- Intro Lesson: c12:09
- Lesson: 111:14
- Lesson: 211:38
- Lesson: 310:14
- Lesson: 49:02
- Lesson: 510:45
Orthogonal projections
Lessons
The Orthogonal Decomposition Theorem
Let S be a subspace in Rn. Then each vector y in Rn can be written as:
y=y^+z
where y^ is in S and z is in S⊥. Note that y^ is the orthogonal projection of y onto S
If {v1,⋯,vp} is an orthogonal basis of S, then
projsy=y^=v1⋅v1y⋅v1v1+v2⋅v2y⋅v2v2+⋯+vp⋅vpy⋅vpvp
However if {v1,⋯,vp} is an orthonormal basis of S, then
projsy=y^=(y⋅v1)v1+(y⋅v2)v2+⋯+(y⋅vp)vp
Property of Orthogonal Projection
If {v1,⋯,vp} is an orthogonal basis for S and if y happens to be in S, then
projsy=y
In other words, if y is in S=Span{v1,⋯,vp}, then projSy=y.
The Best Approximation Theorem
Let S be a subspace of Rn. Also, let y be a vector in Rn, and y^ be the orthogonal projection of y onto S. Then y is the closest point in S, because
∥y−y^∥ < ∥y−u∥
where u are all vectors in S that are distinct from y^.
Let S be a subspace in Rn. Then each vector y in Rn can be written as:
where y^ is in S and z is in S⊥. Note that y^ is the orthogonal projection of y onto S
If {v1,⋯,vp} is an orthogonal basis of S, then
However if {v1,⋯,vp} is an orthonormal basis of S, then
Property of Orthogonal Projection
If {v1,⋯,vp} is an orthogonal basis for S and if y happens to be in S, then
In other words, if y is in S=Span{v1,⋯,vp}, then projSy=y.
The Best Approximation Theorem
Let S be a subspace of Rn. Also, let y be a vector in Rn, and y^ be the orthogonal projection of y onto S. Then y is the closest point in S, because
where u are all vectors in S that are distinct from y^.
- IntroductionOrthogonal Projections Overview:a)The Orthogonal Decomposition Theorem
• Make y as the sum of two vectors y^ and z
• Orthogonal basis → y^=v1⋅v1y⋅v1v1+⋯+vp⋅vpy⋅vpvp
• Orthonormal basis → y^=(y⋅v1)v1+⋯+(y⋯vp)vp
• z=y−y^b)Property of Orthogonal Projections
• projsy=y
• Only works if y is in Sc)The Best Approximation Theorem
• What is the point closest to y in S? y^!
• Reason why: ∥y−y^∥ < ∥y−u∥
• The Distance between the y and y^ - 1.The Orthogonal Decomposition Theorem
Assume that {v1,v2,v3} is an orthogonal basis for Rn. Write y as the sum of two vectors, one in Span{v1}, and one in Span{v2,v3}. You are given that:
- 2.Verify that {v1,v2} is an orthonormal set, and then find the orthogonal projection of y onto Span{v1,v2}.
- 3.Best Approximation
Find the best approximation of y by vectors of the form c1v1+c2v2, where, and
,
.
- 4.Finding the Closest Point and Distance
Find the closest point to y in the subspace S spanned by v1 and v2. - 5.Find the closest distance from y to S=Span{v1,v2} if