Linear Algebra/Orthogonal Projection Onto a Line/Solutions

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This exercise is recommended for all readers.
Problem 1

Project the first vector orthogonally onto the line spanned by the second vector.

  1. ,
  2. ,
  3. ,
  4. ,

Each is a straightforward application of the formula from Definition 1.1.

This exercise is recommended for all readers.
Problem 2

Project the vector orthogonally onto the line.

  1. , the line
  1. Writing the line as gives this projection.
Problem 3

Although the development of Definition 1.1 is guided by the pictures, we are not restricted to spaces that we can draw. In project this vector onto this line.


This exercise is recommended for all readers.
Problem 4

Definition 1.1 uses two vectors and . Consider the transformation of resulting from fixing

and projecting onto the line that is the span of . Apply it to these vectors.

Show that in general the projection tranformation is this.

Express the action of this transformation with a matrix.


In general the projection is this.

The appropriate matrix is this.

Problem 5

Example 1.5 suggests that projection breaks into two parts, and , that are "not interacting". Recall that the two are orthogonal. Show that any two nonzero orthogonal vectors make up a linearly independent set.


Suppose that and are nonzero and orthogonal. Consider the linear relationship . Take the dot product of both sides of the equation with to get that

is equal to . With the assumption that is nonzero, this gives that is zero. Showing that is zero is similar.

Problem 6
  1. What is the orthogonal projection of onto a line if is a member of that line?
  2. Show that if is not a member of the line then the set is linearly independent.
  1. If the vector is in the line then the orthogonal projection is . To verify this by calculation, note that since is in the line we have that for some scalar .
    (Remark. If we assume that is nonzero then the above is simplified on taking to be .)
  2. Write for the projection . Note that, by the assumption that is not in the line, both and are nonzero. Note also that if is zero then we are actually considering the one-element set , and with nonzero, this set is necessarily linearly independent. Therefore, we are left considering the case that is nonzero. Setting up a linear relationship
    leads to the equation . Because isn't in the line, the scalars and must both be zero. The case is handled above, so the remaining case is that , and this gives that also. Hence the set is linearly independent.
Problem 7

Definition 1.1 requires that be nonzero. Why? What is the right definition of the orthogonal projection of a vector onto the (degenerate) line spanned by the zero vector?


If is the zero vector then the expression

contains a division by zero, and so is undefined. As for the right definition, for the projection to lie in the span of the zero vector, it must be defined to be .

Problem 8

Are all vectors the projection of some other vector onto some line?


Any vector in is the projection of some other onto a line, provided that the dimension is greater than one. (Clearly, any vector is the projection of itself into a line containing itself; the question is to produce some vector other than that projects to .)

Suppose that with . If then we consider the line and if we take to be any (nondegenerate) line at all (actually, we needn't distinguish between these two cases— see the prior exercise). Let be the components of ; since , there are at least two. If some is zero then the vector is perpendicular to . If none of the components is zero then the vector whose components are is perpendicular to . In either case, observe that does not equal , and that is the projection of onto .

We can dispose of the remaining and cases. The dimension case is the trivial vector space, here there is only one vector and so it cannot be expressed as the projection of a different vector. In the dimension case there is only one (nondegenerate) line, and every vector is in it, hence every vector is the projection only of itself.

This exercise is recommended for all readers.
Problem 9

Show that the projection of onto the line spanned by has length equal to the absolute value of the number divided by the length of the vector .


The proof is simply a calculation.

Problem 10

Find the formula for the distance from a point to a line.


Because the projection of onto the line spanned by is

the distance squared from the point to the line is this (a vector dotted with itself is written ).

Problem 11

Find the scalar such that is a minimum distance from the point by using calculus (i.e., consider the distance function, set the first derivative equal to zero, and solve). Generalize to .


Because square root is a strictly increasing function, we can minimize instead of the square root of . The derivative is . Setting it equal to zero gives the only critical point.

Now the second derivative with respect to

is strictly positive (as long as neither nor is zero, in which case the question is trivial) and so the critical point is a minimum.

The generalization to is straightforward. Consider , take the derivative, etc.

This exercise is recommended for all readers.
Problem 12

Prove that the orthogonal projection of a vector onto a line is shorter than the vector.


The Cauchy-Schwarz inequality gives that this fraction

when divided by is less than or equal to one. That is, is larger than or equal to the fraction.

This exercise is recommended for all readers.
Problem 13

Show that the definition of orthogonal projection onto a line does not depend on the spanning vector: if is a nonzero multiple of then equals .


Write for , and calculate: .

This exercise is recommended for all readers.
Problem 14

Consider the function mapping to plane to itself that takes a vector to its projection onto the line . These two each show that the map is linear, the first one in a way that is bound to the coordinates (that is, it fixes a basis and then computes) and the second in a way that is more conceptual.

  1. Produce a matrix that describes the function's action.
  2. Show also that this map can be obtained by first rotating everything in the plane radians clockwise, then projecting onto the -axis, and then rotating radians counterclockwise.
  1. Fixing
    as the vector whose span is the line, the formula gives this action,
    which is the effect of this matrix.
  2. Rotating the entire plane radians clockwise brings the line to lie on the -axis. Now projecting and then rotating back has the desired effect.
Problem 15

For let be the projection of onto the line spanned by , let be the projection of onto the line spanned by , let be the projection of onto the line spanned by , etc., back and forth between the spans of and . That is, is the projection of onto the span of if is even, and onto the span of if is odd. Must that sequence of vectors eventually settle down— must there be a sufficiently large such that equals and equals ? If so, what is the earliest such ?


The sequence need not settle down. With

the projections are these.

This sequence doesn't repeat.

Linalg projection sequence.png