2.9.1 Lab 5¶
Part I. Find a basis for the span of the set of vectors¶
For the set of vectors indicated, find a basis for their span.
If your birthday is in:
January through April, do the sets of vectors \(\vec v\).
May through August, do the sets of vectors \(\vec x\).
September through December, do the sets of vectors \(\vec y\).
\[\begin{split}\vec v_1 =\left[\begin{array}{r}0\\0\\0\\1\\\end{array}\right],\vec v_2=\left[\begin{array}{r}0\\-1\\1\\1\\\end{array}\right],\vec v_3=\left[\begin{array}{r}0\\3\\-3\\4\\\end{array}\right],\vec v_4=\left[\begin{array}{r}-1\\2\\-4\\4\\\end{array}\right]\end{split}\]
v1 = [0 ; 0 ; 0 ; 1 ];
v2 = [0 ; -1 ; 1 ; 1 ];
v3 = [0 ; 3 ; -3 ; 4 ];
v4 = [-1 ; 2 ; -4 ; 4 ];
\[\begin{split}\vec x_1 =\left[\begin{array}{r}4\\-4\\0\\0\end{array}\right],\vec x_2=\left[\begin{array}{r}1\\-1\\0\\0\end{array}\right],\vec x_3=\left[\begin{array}{r}1\\-5\\0\\0\end{array}\right],\vec x_4=\left[\begin{array}{r}-1\\21\\1\\1\end{array}\right]\end{split}\]
x1 = [4 ; -4 ; 0 ; 0 ];
x2 = [1 ; -1 ; 0 ; 0 ];
x3 = [1 ; -5 ; 0 ; 0 ];
x4 = [-1 ; 21 ; 1 ; 1 ];
\[\begin{split}\vec y_1 =\left[\begin{array}{r}-4\\0\\0\\-4\\\end{array}\right],\vec y_2=\left[\begin{array}{r}2\\3\\0\\2\\\end{array}\right],\vec y_3=\left[\begin{array}{r}6\\-2\\0\\6\\\end{array}\right],\vec y_4=\left[\begin{array}{r}19\\5\\1\\19\\\end{array}\right]\end{split}\]
y1 = [-4 ; 0 ; 0 ; -4 ];
y2 = [2 ; 3 ; 0 ; 2 ];
y3 = [6 ; -2 ; 0 ; 6 ];
y4 = [19 ; 5 ; 1 ; 19 ];
Part II. Find a basis for the column space of the matrix¶
For the matrix indicated, find a basis for the column space.
If your first name begins with:
A through D, do A.
D through J, do D.
J through Z, do J.
\[\begin{split}A = \left[\begin{array}{rrr}1&23&8\\1&18&5\\0&0&1\\\end{array}\right]\end{split}\]
A = [1 23 8 ; 1 18 5 ; 0 0 1 ];
\[\begin{split} D = \left[\begin{array}{rrr}1&10&2\\0&0&1\\0&1&-2\\\end{array}\right]\end{split}\]
D = [1 10 2 ; 0 0 1 ; 0 1 -2 ];
\[\begin{split}J = \left[\begin{array}{rrr}-6&5&-12\\0&1&1\\8&-6&17\\\end{array}\right]\end{split}\]
J = [-6 5 -12 ; 0 1 1 ; 8 -6 17 ];
Part III: Find the \(\mathcal B\)-coordinates of the vector¶
Write the vector
\[\begin{split}\vec w = \left[\begin{array}{r}8\\-2\\\end{array}\right]\end{split}\]
in \(\mathcal B\)-coordinates where the basis vectors \(\vec b_1,\vec b_2\) are given.
If your last name begins with:
B through G, do Set 1.
H through Q, Set 2.
R through Z, Set 3.
A pick any of the three.
Set 1
\[\begin{split} \vec b_1 =\left[\begin{array}{r}3\\1\end{array}\right],\vec b_2=\left[\begin{array}{r}4\\1\\\end{array}\right]\end{split}\]
B = [3 4 ; 1 1 ];
Set 2
\[\begin{split} \vec b_1 =\left[\begin{array}{r}-3\\3\\\end{array}\right],\vec b_2=\left[\begin{array}{r}-7\\2\\\end{array}\right] \end{split}\]
B = [-3 -7 ; 3 2 ];
Set 3
\[\begin{split}\vec b_1 =\left[\begin{array}{r}-8\\6\\\end{array}\right],\vec b_2=\left[\begin{array}{r}3\\-1\\\end{array}\right]\end{split}\]
B= [-8 3 ; 6 -1 ];