6.3 - true/false

The best approximation to y by elements of a subspace W is given by the vector y - projwy.

False - The best approximation to y by elements of a subspace W is projwy.

The orthogonal projection yhat of y onto a subspace W can sometimes depend on the orthogonal basis for W used to compute yhat.

False - The orthogonal projection that of y onto a subspace W is always dependent on W only, not on any particular basis.

If an n x p matrix U has orthonormal columns, then UU^Tx = x for all x in Rn.

False - This is only true if U is square, which the statement does not indicate.

In the Orthogonal Decomposition Theorem each term in formula (2) for yhat is itself an orthogonal projection of y onto a subspace of W.

True - Each term in projwy = (y . ui / ui . ui )ui +.... + (y . up / up . up )up is itself an orthogonal projection of y onto a 1 dimensional subspace spanned by one of the u's in W's basis.

If the columns of an n x p matrix U are orthonormal, then UU^Ty is the orthogonal projection of y onto the column space of U.

True - UU^Ty = projwy = (y . u1)u1 + ..... (y . un)un, where u1, ..., un are orthonormal.

For each y and each subspace W, the vector y - projwy is orthogonal to W.

True - by the Orthogonal Decomposition Theorem - for the vector subspace W of Rn, each y in Rn can be written as y = projwy + z and z = y - projwy (which is an element of W complement) since projwy = Summation_from_i=1_to_n((y . ui / ui . ui ) times ui). and this summation is an element in W because every part of the summation is an element in W. y-projwy is the orthogonal complement, which makes the statement true.

If W is a subspace of Rn and if v is in both W and W complement, then v must be the zero vector.

True - p 350 - bottom example

If y = z1 + z2, where z1 is in a subspace W and z2 is in W complement, then z1 must be the orthogonal projection of y onto W.

True - p 350 - bottom example

If y is in a subspace W, then the orthogonal projection of y onto W is y itself.

True - p 352 - "If y is in W = Span{u1, ..., up}, then projwy = y."

If z is orthogonal to u1 and to u2 and if W = Span{u1, u2}, then z must be in W complement.

True - z is orthogonal to any linear combination of u1 and u2