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*}
(a)
Find a basis for the kernel of \(T\) (that is, the nullspace of \(A\)).
(b)
Extend the basis from part (a) to a basis of \(\R^7\text{.}\)
Suggestion: recall that you can extend to a basis by adding elements of the standard basis for \(\R^7\text{.}\)
Use the code cell below to enter a matrix you can use to confirm that your basis is, in fact, a basis.
Once your testing indicates that you have successfully found a basis, input your basis vectors below.
Suggestion: in part (a), you probably used the command
A.nullspace() to find the basis for \(\ker(T)\text{.}\) If you enter this asB1 = 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..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.)(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.
(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.
Note: you can compute the dot product of vectors
X and Y using the syntax X.dot(Y).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.
(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.
