Project: Orthogonal bases

Section 3.4 Project: Orthogonal bases

This worksheet will give you an opportunity to work with orthogonal sets of vectors in \(\R^n\text{,}\) and explore some of the related results. First, we will look at the construction of an orthogonal complement.

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Exercise 3.4.1.

Let \(T:\R^7\to\R^4\) be given by \(T(\xx)=A\xx\text{,}\) where

\begin{equation*} A = \bbm 3\amp -4\amp 2\amp 0 \amp 1\amp 7\amp -5\\ -4\amp 7\amp 0\amp -3\amp 2 \amp 0\amp 1\\ 0\amp 5\amp 8\amp -9\amp 10\amp 27\amp -19\\ 2\amp 7\amp -3\amp 1\amp 5\amp 12\amp -9\ebm\text{.} \end{equation*}

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(a)

Find a basis for the kernel of \(T\) (that is, the nullspace of \(A\)).

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(b)

Extend the basis from part (a) to a basis of \(\R^7\text{.}\)

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Suggestion: recall that you can extend to a basis by adding elements of the standard basis for \(\R^7\text{.}\)

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Use the code cell below to enter a matrix you can use to confirm that your basis is, in fact, a basis.

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Once your testing indicates that you have successfully found a basis, input your basis vectors below.

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Suggestion: in part (a), you probably used the command A.nullspace() to find the basis for \(\ker(T)\text{.}\) If you enter this as

B1 = A.nullspace()
B1

then you can use the name B1 to recall those vectors, so you don’t have to input them manually. The object B1 is a list of vectors, and you can call elements of that list as B1[0], B1[1], etc..

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First, input the vectors that you added to B1 to get a basis of \(\R^7\text{.}\) (For example, you might enter e1 = Matrix([1,0,0,0,0,0,0]).) Then define your basis as a list B. For example, you might enter B = (B1[0],B1[1],B1[2],e1,e2,e3,e4), if you found that there were three vectors in the list B1, and that the first 4 standard basis vectors were sufficient to get you a basis for \(\R^7\text{.}\) (This may not be what you find, of course.)

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(c)

Use the Gram-Schmidt algorithm to create an orthogonal basis for \(\R^7\) from the basis \(B\text{.}\) You will want to give your basis a new name, so that you can reference its elements later.

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(d)

Let \(U = \ker(T)\text{.}\) If \(\dim U = k\text{,}\) then the first \(k\) vectors of your orthogonal basis from \(\R^7\) form an orthogonal basis for \(U\text{,}\) according to the Gram-Schmidt theorem. Confirm that each of the remaining vectors in this basis are orthogonal to the vectors in the basis B1.

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Note: you can compute the dot product of vectors X and Y using the syntax X.dot(Y).

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It follows that the remaining vectors are elements of \(U^\bot\text{.}\) In fact, they form a basis. we know they are independent, because they are part of a basis for \(\R^7\text{.}\) And since \(\dim U+\dim U^\bot = \dim \R^7\text{,}\) we know we have the right number.

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(e)

Given the vector \(\xx = (4,-1,6,3,8,2,5)\in\R^7\text{,}\) find a vector \(\mathbf{p}\in U=\ker(T)\) such that \(\len{\xx-\mathbf{p}}\) is as small as possible.

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Exercise 3.4.2.

Theorem 3.3.10 states the following: given a subspace \(U\subseteq \R^n\) with basis \(B = \{\uu_1,\ldots,\uu_k\}\) (not necessarily an orthogonal basis), let \(A\) be the \(n\times k\) matrix whose columns are the vectors in \(B\text{.}\) Then \(U^\bot = \nll(A^T)\text{.}\)

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(a)

Put another way, this theorem states that \(\xx\in U^\bot\) if and only if \(A^T\xx = \zer\text{.}\) Give a proof (or at least, a convincing explanation) of this fact.

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(b)

Let \(U = \{(4a-3b+2c, a-7b+5c,9a-5c,-3b-7c,-3a+2b-5c,4a-7b)\}\in \R^6\text{.}\) Determine a basis for \(U\text{,}\) and input the vectors below.

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(c)

Use Theorem 3.3.11 to find a basis for \(U^\bot\text{.}\)

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