3 V Projection 3 V 1 Orthogonal Projection

3. V. Projection 3. V. 1. Orthogonal Projection Into a Line 3. V. 2. Gram-Schmidt Orthogonalization 3. V. 3. Projection Into a Subspace 3. V. 1. & 2. deal only with inner product spaces. 3. V. 3 is applicable to any direct sum space.

3. V. 1. Orthogonal Projection Into a Line Definition 1. 1: Orthogonal Projection The orthogonal projection of v into the line spanned by a nonzero s is the vector. Example 1. 3: Orthogonal projection of the vector ( 2 3 )T into the line y = 2 x.

Example 1. 4: Orthogonal projection of a general vector in R 3 into the y-axis Example 1. 5: Project = Discard orthogonal components A railroad car left on an east-west track without its brake is pushed by a wind blowing toward the northeast at fifteen miles per hour; what speed will the car reach?

Example 1. 6: Nearest Point A submarine is tracking a ship moving along the line y = 3 x + 2. Torpedo range is one-half mile. Can the sub stay where it is, at the origin on the chart, or must it move to reach a place where the ship will pass within range? Ship’s path is parallel to vector Point p of closest approach is (Out of range)

Exercises 3. V. 1. 1. Consider the function mapping a plane to itself that takes a vector to its projection into the line y = x. (a) Produce a matrix that describes the function’s action. (b) Show also that this map can be obtained by first rotating everything in the plane π/4 radians clockwise, then projecting into the x-axis, and then rotating π/4 radians counterclockwise.

3. V. 2. Gram-Schmidt Orthogonalization Given a vector s, any vector v in an inner product space can be decomposed as where Definition 2. 1: Mutually Orthogonal Vectors v 1, …, vk Rn are mutually orthogonal if vi · vj = 0 i j Theorem 2. 2: A set of mutually orthogonal non-zero vectors is linearly independent. Proof: → → cj = 0 j Corollary 2. 3: A set of k mutually orthogonal nonzero vectors in V k is a basis for the space. Definition 2. 5: Orthogonal Basis An orthogonal basis for a vector space is a basis of mutually orthogonal vectors.

Example 2. 6: Turn into an orthogonal basis for R 3.

Theorem 2. 7: Gram-Schmidt Orthogonalization If β 1 , …, βk is a basis for a subspace of Rn the κj s calculated from the following scheme is an orthogonal basis for the same subspace. Proof: Then For m 2 , Let β 1 , …, βm be mutually orthogonal, and QED

If each κj is furthermore normalized, the basis is called orthonormal. The Gram-Schmidt scheme simplifies to:

Exercises 3. V. 2. 1. Perform the Gram-Schmidt process on this basis for R 3, 2. Show that the columns of an n n matrix form an orthonormal set if and only if the inverse of the matrix is its transpose. Produce such a matrix.

3. V. 3. Projection Into a Subspace Definition 3. 1: For any direct sum V = M N and any v V such that v=m+n with m M and n N The projection of v into M along N is defined as proj. M, N (v) = m Reminder: • M & N need not be orthogonal. • There need not even be an inner product defined.

Example 3. 2: The space M 2 2 of 2 2 matrices is the direct sum of Task: Find proj. M , N (A), where Solution: Let the bases for M & N be → ∴ is a basis for M 2 2.

Example 3. 3: Both subscripts on proj. M , N (v) are significant. Consider with basis and subspaces & It’s straightforward to verify Task: Find proj. M , N (v) and proj. M , L (v) where Solution: For →

For → Note: BML is orthogonal but BMN is not.

Definition 3. 4: Orthogonal Complement The orthogonal complement of a subspace M of Rn is M = { v Rn | v is perpendicular to all vectors in M } ( read “M perp” ). The orthogonal projection proj. M (v ) of a vector is its projection into M along M . Example 3. 5: In R 3, find the orthogonal complement of the plane Solution: → Natural basis for P is ( parameter = z)

Lemma 3. 7: Let M be a subspace of Rn. Hence, v Rn, Then M is also a subspace and Rn = M M . v proj. M (v) is perpendicular to all vectors in M. Proof: Construct bases using G-S orthogonalization. Theorem 3. 8: Let v be a vector in Rn and let M be a subspace of Rn with basis β 1 , …, βk . If A is the matrix whose columns are the β’s then proj. M (v ) = c 1β 1 + …+ ck βk where the coefficients ci are the entries of the vector (AT A) AT v. That is, proj. M (v ) = A (AT A) 1 AT v. Proof: By lemma 3. 7, where c is a column vector → → →

Interpretation of Theorem 3. 8: If B = β 1 , …, βk is an orthonormal basis, then ATA = I. In which case, proj. M (v ) = A (AT A) 1 AT v = A AT v. with In particular, if B = Ek , then A = AT = I. In case B is not orthonormal, the task is to find C s. t. B = AC and BTB = I. → Hence →

Example 3. 9: To orthogonally project From → into subspace we get

Exercises 3. V. 3. 1. Project v into M along N, where 2. Find M for

3. Define a projection to be a linear transformation t : V → V with the property that repeating the projection does nothing more than does the projection alone: ( t t )(v) = t (v) for all v V. (a) Show that for any such t there is a basis B = β 1 , …, βn for V such that where r is the rank of t. (b) Conclude that every projection has a block partial-identity representation: