recognize the method for projecting a vector onto a subspace, then
consider how to find the best approximation of a vector in a subspace using projection.
Subsection6.3.1Projection
Instructions.
Drag point C around. See how the orthogonal projection (shadow from the sun) is affected.
Figure6.3.1.Illustration of Orthogonal Projection
Projection matches the concept of a shadow being formed by a light. Orthogonal projection considers the case where the light is above the object making the shadow on a line orthogonal to the line on which the shadow is formed. See Figure 6.3.1 for an example. Note \(\vec{w}\) is the projection of \(\vec{v}\) onto \(\vec{u}\text{.}\)
Activity6.3.1.
In this activity we consider what causes differences in a projection of one vector onto another.
(a)
Move point C to the right. Note this changes the angle and length of \(\vec{v}\text{.}\) How does the projection \(\vec{w}\) change as a result?
(b)
Move point C up. How does the projection \(\vec{w}\) change as a result?
(c)
What effect does the length of \(\vec{u}\) have on the projection?
(d)
Move point C to the left. Continue until past where the vectors meet. How does the projection \(\vec{w}\) compare to the vector \(\vec{u}\) in this case?
(e)
On what is the direction of the projection \(\vec{w}\) based? Consider \(\vec{v}\) and \(\vec{u}\text{.}\)
Definition6.3.2.Projection.
The projection of \(\vec{u}\) onto \(\vec{v}\) is \(\proj_{\vec{v}} \vec{u} = \frac{\langle \vec{u}, \vec{v} \rangle}{\langle \vec{v}, \vec{v} \rangle} \vec{v}\text{.}\)
Checkpoint6.3.3.
Calculate the projection of the first vector onto the second.
(a)
\([1,2,3]\) onto \([1,0,0]\)
(b)
\([1,2,3]\) onto \([2,0,0]\)
(c)
\([1,2,3]\) onto \([-1,0,0]\)
(d)
\([1,2,3]\) onto \([1,1,1]\)
(e)
\([1,1,1]\) onto \([1,2,3]\)
Subsection6.3.2Projection onto a Space
Figure6.3.4.Sundial showing arm and shadow
The following observations based on a sundial illustrate concepts of projection.
Note that \(\proj_{\vec{u}} \vec{v}\) projects a 1D vector onto a 1D line (extended vector).
A light projects a shadow of a 1D (effectively) needle onto a 2D surface.
For the following experiment let the origin be at the intersection of the needle with the surface.
Figure6.3.5.Projection of a Vector onto a Plane
Activity6.3.2.
In this activity we calculate projections onto two vectors and consider where the shadow (projection) onto the plane defined by these two vectors would end up.
Use the vectors \(\vec{n}=[2,1,1]\text{,}\)\(\B=\{ [1,0,0], [0,1,0] \}\text{.}\)
(a)
Calculate \(\proj_{[1,0,0]} \vec{n}\text{.}\)
(b)
Calculate \(\proj_{[0,1,0]} \vec{n}\text{.}\)
(c)
Sketch these two projections on the image.
(d)
Noting that these projections suppose the light source is directly above the origin, sketch where you expect the shadow of \(\vec{n}\) would appear.
(e)
Compare this shadow to the two projections onto the vectors.
(f)
Based on this, what definition for projection onto a subspace makes sense?
Activity6.3.3.
The first part of this activity applies the concept from above to an illustrative game. The second part of this activity introduces an idea for approximation.
You control Lucky the leaping lizard using your keyboard. The arrow keys move you in the directions \((1,0,0),\)\((-1,0,0),\)\((0,1,0),\)\((0,-1,0)\) and the space bar causes Lucky to jump \((0,0,1)\text{.}\)
(a)
How do you reach \((3,2,1)\text{?}\)
(b)
How do you reach \((-2,7,1)\text{?}\)
(c)
What is the relationship between the set of arrow key vectors and the jump vector?
(d)
Given a subspace \(W\) of \(\R^n\) how can you reach any vector \(\vec{x} \in \R^n\) using one vector in \(W\) and one vector in \(W^\perp\text{?}\)
(e)
How is the one vector in \(W\) calculated?
(f)
Suppose the space bar is broken. What is the closest you can get to \((3,2,1)\text{?}\)
(g)
How can this location be calculated using recent ideas?
Theorem6.3.6.Orthogonal Decomposition Theorem.
For any subspace \(W\) of a vector space \(X\) any vector \(\vec{x} \in X\) can be written uniquely in the form
where \(\vec{w} \in W\) and \(\vec{z} \in W^{\perp}\text{.}\)
Subsection6.3.3Approximation Theorem
Activity6.3.4.
This activity draws our attention to details that can prove our guess above about the closest point is correct.
Consider \(\vec{y}=[8,1,7]\text{,}\)\(W\) is the span of \(\{[1,0,0],[0,1,0]\}\text{,}\) and \(\vec{v}=[1,3,0]\text{.}\)
(a)
Draw \(\vec{y},\)\(\hat{y} = \proj_W \vec{y}\text{,}\) and \(\vec{v}\text{.}\) Draw them as points not arrows.
(b)
What figure do the points produce?
(c)
Because \(\hat{y}\) is a projection what special type of figure is this?
(d)
Find the distances \(\|\vec{y}-\hat{y}\|\) and \(\|\vec{y}-\vec{v}\|\text{.}\) Note \(\vec{v}\) is also in \(W\text{.}\) You may wish to apply the Pythagoran theorem.
(e)
Compare the distance between \(\vec{y}\) and \(\hat{y}\) to the distance between \(\vec{y}\) and \(\vec{v}\text{.}\)
(f)
What would be the change if we selected any other (than \(\vec{v})\) point in \(W\text{?}\)
(g)
As a result, what is the closest point to \(\vec{y}\) in \(W\text{?}\)
Theorem6.3.7.Orthogonal Approximation.
For any subspace \(W\) of a vector space \(X\text{,}\)\(\|\vec{y}-\hat{y}\| \le \|\vec{y}-\vec{v}\|\) where \(\hat{y}\) is the projection of \(\vec{y}\) onto \(W\) and \(\vec{v}\) is any other vector in \(W\text{.}\)
