DLA Discovery guide

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Discovery guide 37.1 Discovery guide

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Recall that two vectors

$\uvec{u},\uvec{v}$ in

$\R^n$ are called orthogonal if

$\udotprod{u}{v} = 0\text{.}$

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In analogy with this, we will also call two vectors

$\uvec{u},\uvec{v}$ in an inner product space orthogonal if

$\uvecinprod{u}{v} = 0\text{.}$

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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

$U\text{.}$

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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.

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(a)

Describe $\orthogcmp{U}$ if $U$ is a plane through the origin.

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(b)

Describe $\orthogcmp{W}$ if $W$ is a line through the origin.

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(c)

Based on your two answers, make a general conjecture about $\orthogcmp{(\orthogcmp{U})}$ in an inner product space.

Note: In Task a and Task b keep in mind that changing the initial point of a geometric vector doesn't change the vector.

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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 $\uvec{u}_1,\uvec{u}_2,\uvec{u}_3\text{.}$

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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

$\orthogcmp{U}\text{.}$

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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.

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(a)

Determine a basis for $U\text{.}$

Hint

Describe a typical element in $U$ using parameters, then associate a basis vector to each independent parameter.

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(b)

Determine a basis for $\orthogcmp{U}\text{.}$

Hint

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.

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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.

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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.

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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

$\begin{gather} k_1 \uvec{v}_1 + k_2 \uvec{v}_2 + k_3 \uvec{v}_3 = \zerovec\text{.}\label{equation-inprod-orthog-discovery-indep}\tag{$\star$} \end{gather}$

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(a)

From our initial equation ( $\star$ ), we have

$\begin{equation*} \inprod{k_1 \uvec{v}_1 + k_2 \uvec{v}_2 + k_3 \uvec{v}_3}{\uvec{v}_1} = \inprod{\zerovec}{\uvec{v}_1}\text{.} \end{equation*}$

Simplify the left-hand side of this equality to discover something about $k_1\text{.}$

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(b)

Convince yourself that similar reasoning will work for $k_2,k_3\text{.}$

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(c)

Is $\{ \uvec{v}_1, \uvec{v}_2, \uvec{v}_3 \}$ an independent set?

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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

$\begin{gather} \uvec{v} = k_1 \uvec{e}_1 + k_2 \uvec{e}_2 + k_3 \uvec{e}_3\label{equation-inprod-orthog-discovery-3dim-expansion}\tag{$\star\star$} \end{gather}$

for some scalars $k_1,k_2,k_3\text{.}$

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(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 $k_1\text{.}$

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(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 $k_3\text{.}$

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(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 .

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(d)

How does your pattern from Task c change for an orthonormal basis?

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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.

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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.

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(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?

Hint

We already considered this sort of question in Discovery 37.5 above.

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(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 $\uvec{e}_1\text{.}$

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(c)

Carry out a similar exploration process as in Task a and Task b, but for $\uvec{v}_3\text{.}$

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(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 $k_2\text{?}$

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(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 $\uvec{e}_2\text{.}$

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(d)

Extract the pattern: If $V$ has dimension $n$ (instead of dimension $3$ ), then the next step in the procedure would be

$\begin{equation*} \uvec{e}_4 = \underline{\hspace{9.090909090909092em}} \text{.} \end{equation*}$

And then

$\begin{equation*} \uvec{e}_5 = \underline{\hspace{9.090909090909092em}} \text{.} \end{equation*}$

And so on.

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(e)

At the end of all this, what would you do if you wanted an orthonormal basis for $V\text{?}$

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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?

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Discovery 37.8.

Suppose $U$ is a subspace of an inner product space $V\text{.}$

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(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 $U$ ?

Hint

Proposition 17.5.2.

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(b)

Every basis for $U$ can be enlarged to a basis for $V$ (Statement 1 of Proposition 20.5.8).

Suppose

$\begin{equation*} \basisfont{B}_0 = \{ \uvec{u}_1, \dotsc, \uvec{u}_m, \uvec{v}_1, \dotsc, \uvec{v}_\ell \} \end{equation*}$

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

$\begin{equation*} \basisfont{B} = \{ \uvec{e}_1, \dotsc, \uvec{e}_m, \uvec{f}_1, \dotsc, \uvec{f}_\ell \} \end{equation*}$

for $V\text{,}$ what subspace do the $\uvec{f}_j$ span?