Discovery guide 37.1 Discovery guide
Recall that two vectors \(\uvec{u},\uvec{v}\) in \(\R^n\) are called orthogonal if \(\udotprod{u}{v} = 0\text{.}\)
In analogy with this, we will also call two vectors \(\uvec{u},\uvec{v}\) in an inner product space orthogonal if \(\uvecinprod{u}{v} = 0\text{.}\)
Suppose \(V\) in an inner product space, and \(U\) is a subspace of \(V\text{.}\) The collection of all vectors orthogonal to \(U\) is called the orthogonal complement of \(U\text{,}\) and is denoted \(\orthogcmp{U}\text{.}\) That is, \(\orthogcmp{U}\) consists of all vectors that are orthogonal to every vector in \(U\text{.}\)
Discovery 37.1.
Suppose \(V = \R^3\) with the usual Euclidean inner product (i.e. the dot product). Then orthogonal is the same as perpendicular.
(a)
Describe \(\orthogcmp{U}\) if \(U\) is a plane through the origin.
(b)
Describe \(\orthogcmp{W}\) if \(W\) is a line through the origin.
(c)
Based on your two answers, make a general conjecture about \(\orthogcmp{(\orthogcmp{U})}\) in an inner product space.
Discovery 37.2.
Suppose \(U = \Span \{\uvec{u}_1,\uvec{u}_2,\uvec{u}_3\}\) is a subspace of an inner product space \(V\text{.}\) Convince yourself that a vector \(\uvec{v}\) is in \(\orthogcmp{U}\) if and only if \(\uvec{v}\) is orthogonal to each of \(\uvec{u}_1,\uvec{u}_2,\uvec{u}_3\text{.}\)
Since \(\orthogcmp{U}\) is defined by a homogeneous condition (inner product equals \(0\)), we expect it to be a subspace. The orthogonality condition can be used to determine a basis for \(\orthogcmp{U}\text{.}\)
Discovery 37.3.
Consider \(V = \matrixring_{2 \times 2}(\R)\) as an inner product space with \(\inprod{A}{B} = \trace(\utrans{B} A)\text{.}\) Let \(U\) represent the subspace of every upper triangular matrix whose upper-right entry is equal to its trace.
(a)
Determine a basis for \(U\text{.}\)
Describe a typical element in \(U\) using parameters, then associate a basis vector to each independent parameter.
(b)
Determine a basis for \(\orthogcmp{U}\text{.}\)
Use the idea of Discovery 37.2 applied to your basis for \(U\) from Task a to set up a homogeneous system of equations to solve.
A set of vectors in an inner product space is called an orthogonal set if each vector in the set is orthogonal to every other vector in the set. A set of vectors is called an orthonormal set if it is an orthogonal set where every member is a unit vector.
Geometrically we think of linearly independent vectors as “pointing in different directions,” so it is reasonable to expect an orthogonal set of vectors to be independent.
Discovery 37.4.
Suppose \(\{ \uvec{v}_1, \uvec{v}_2, \uvec{v}_3 \}\) is an orthogonal set of nonzero vectors in an inner product space. To test for independence, we start with the homogeneous vector equation
(a)
From our initial equation (\(\star\)), we have
Simplify the left-hand side of this equality to discover something about \(k_1\text{.}\)
(b)
Convince yourself that similar reasoning will work for \(k_2,k_3\text{.}\)
(c)
Is \(\{ \uvec{v}_1, \uvec{v}_2, \uvec{v}_3 \}\) an independent set?
Discovery 37.5.
Suppose \(\basisfont{B} = \{\uvec{e}_1,\uvec{e}_2,\uvec{e}_3\}\) is both a basis and an orthogonal set in an inner product space \(V\text{.}\) Since \(\basisfont{B}\) is a basis, every vector \(\uvec{v}\) in \(V\) has a unique expression
for some scalars \(k_1,k_2,k_3\text{.}\)
(a)
Substitute (\(\star\star\)) into \(\inprod{\uvec{v}}{\uvec{e}_1}\) to obtain an expression for \(\inprod{\uvec{v}}{\uvec{e}_1}\) in terms of the \(k_j\) and \(\uvec{e}_j\text{.}\) Then isolate \(k_1\text{.}\)
(b)
Similar to Task a, use (\(\star\star\)) in \(\inprod{\uvec{v}}{\uvec{e}_2}\) to obtain a formula for \(k_2\text{,}\) and in \(\inprod{\uvec{v}}{\uvec{e}_3}\) to obtain a formula for \(k_3\text{.}\)
(c)
Extract the pattern: If \(V\) has dimension \(n\) (instead of dimension \(3\)), then the coordinates of a vector \(\uvec{v}\) relative to an orthogonal basis \(\basisfont{B} = \{\uvec{e}_1,\uvec{e}_2,\dotsc,\uvec{e}_n\}\) are .
(d)
How does your pattern from Task c change for an orthonormal basis?
A basic problem in an inner product space is how to come up with an orthogonal basis. So let's invent a procedure for doing so.
Discovery 37.6.
To keep it simple, let's suppose \(V\) has dimension \(3\text{.}\) The beginning ingredient for our procedure is some (probably non-orthogonal) basis \(\basisfont{B}_0 = \{\uvec{v}_1,\uvec{v}_2,\uvec{v}_3\}\text{,}\) and the end result should be some definitely orthogonal basis \(\basisfont{B} = \{\uvec{e}_1,\uvec{e}_2,\uvec{e}_3\}\text{.}\)
To get the process started, we might as well first choose \(\uvec{e}_1 = \uvec{v}_1\text{,}\) since we don't yet have any other \(\uvec{e}_j\) vectors chosen yet to which \(\uvec{e}_1\) needs to be orthogonal.
The rest of the activity requires us to choose \(\uvec{e}_2,\uvec{e}_3\) to complete the orthogonal basis.
(a)
If we already knew the answer \(\basisfont{B} = \{\uvec{e}_1,\uvec{e}_2,\uvec{e}_3\}\text{,}\) and we expanded \(\uvec{v}_2 = k_1\uvec{e}_1 + k_2 \uvec{e}_2 + k_3 \uvec{e}_3\) relative to \(\basisfont{B}\text{,}\) what would the coefficient \(k_1\) be?
We already considered this sort of question in Discovery 37.5 above.
(b)
Draw a diagram of \(\uvec{v}_2\text{,}\) \(\uvec{e}_1\text{,}\) and \(k_1 \uvec{e}_1\) as if these were vectors in \(\R^n\text{,}\) keeping in mind that \(k_1 \uvec{e}_1\) should be exactly that part of \(\uvec{v}_2\) that is parallel to \(\uvec{e}_1\text{.}\) (Does this diagram remind you of some previous concept?)
Use your diagram to propose a choice of vector \(\uvec{e}_2\) that is orthogonal to \(\uvec{e}_1\text{.}\)
(c)
(i)
For expansion \(\uvec{v}_3 = k_1\uvec{e}_1 + k_2\uvec{e}_2 + k_3\uvec{e}_3\) relative to \(\basisfont{B}\text{,}\) what would be the coordinates \(k_1\) and \(k_2\text{?}\)
(ii)
Draw a diagram of \(\uvec{v}_3\text{,}\) \(\uvec{e}_1\text{,}\) \(\uvec{e}_2\text{,}\) and \(k_1 \uvec{e}_1 + k_2\uvec{e}_2\text{,}\) keeping in mind that \(k_1 \uvec{e}_1 + k_2\uvec{e}_2\) should be exactly that part of \(\uvec{v}_3\) that is “parallel” to \(\Span\{\uvec{e}_1,\uvec{e}_2\}\text{.}\)
Use your diagram to propose a choice of vector \(\uvec{e}_3\) that is orthogonal to both \(\uvec{e}_1\) and \(\uvec{e}_2\text{.}\)
(d)
Extract the pattern: If \(V\) has dimension \(n\) (instead of dimension \(3\)), then the next step in the procedure would be
And then
And so on.
(e)
At the end of all this, what would you do if you wanted an orthonormal basis for \(V\text{?}\)
Discovery 37.7.
What do you think the result would be if you unknowingly applied the procedure of Discovery 37.6 to a starting basis \(\basisfont{B}_0\) that was already orthogonal?
Discovery 37.8.
Suppose \(U\) is a subspace of an inner product space \(V\text{.}\)
(a)
Applying the procedure of Discovery 37.6 to a basis for \(U\) should produce an orthogonal set. But will it produce an orthogonal basis for \(U\)?
(b)
Every basis for \(U\) can be enlarged to a basis for \(V\) (Statement 1 of Proposition 20.5.8).
Suppose
is such an enlarged basis for \(V\text{,}\) so that the \(\uvec{u}_j\) form a basis for \(U\text{.}\) If we apply the procedure of Discovery 37.6 to \(\basisfont{B}_0\) to obtain orthgonal basis
for \(V\text{,}\) what subspace do the \(\uvec{f}_j\) span?