Lab Project 2 (Due 3/14/03)

Date: MATH 2270-1 - Spring 2003

From the course webpage download the file proj2data.m. Load it (this is explained in the third Maple tutorial, also available at the course webpage) into Maple and you will have defined a few matrices and vectors that will be used throughout this project. The naming convention is as follows: the vector $v_1$ in this note becomes the vector vv1 in Maple, and, similarly the matrix $A$ becomes AA.

1. Let $S_1 = <\left( \begin{array}{c} 2\\ 0\\ 1 \end{array}\right) , \left( \begin{array}{c} -1\\ 1 \\ 1 \end{array}\right) >$ and $S_2 = \{ \left( \begin{array}{c} x\\ y\\ z \end{array}\right) \in \ensuremath{\mathbb{R}}^3 : x-y+3z=0\}$.
   1. Find a basis of $S_1$. What is the dimension of $S_1$?
   2. Produce a graph of $S_1$ in $\ensuremath{\mathbb{R}}^3$; the same graph must show $S_1$ as well as the basic vectors you found in the previous point. From the graph, is $S_1$ a line, plane, etc?
   3. Find a basis of $S_2$. What is the dimension of $S_2$?
   4. Produce a graph of $S_2$ in $\ensuremath{\mathbb{R}}^3$; the same graph must show $S_2$ as well as the basic vectors you found in the previous point. From the graph, is $S_2$ a line, plane, etc?
   5. Produce a joint graph of $S_1$ and $S_2$. From the graph, what is the dimension of $S_1\cap S_2$?
   6. Find an expression of $S_1$ in terms of a linear equation (similar to the way in which $S_2$ was introduced above).
   7. Find $S_1\cap S_2$ algebraically. Does your answer agree with the graph that you obtained in 1e.
   8. Explore the command IntersectionBasis from Maple's LinearAlgebra package. Use this command to find a basis for $S_1\cap S_2$.

2. 1. Find a basis $B_1$ and the dimension of

      $$S = <v_1, v_2, v_3, v_4, v_5> \subset \ensuremath{\mathbb{R}}^{10}.$$

      
      
      These vectors are called vv1, etc. in Maple.
   2. Check that the vector $w$ (ww in Maple) is in the subspace $S$ and find the coordinates of $w$ with respect to the basis $B_1$.
   3. Find another (different!) basis $B_2$ of $S$. Write the coordinates of $w$ with respect to $B_2$.
   4. Write the change of bases matrix $C(B_1,B_2)$ and verify the relation with the coordinates of $w$ with respect to the bases $B_1$ and $B_2$.

3. 1. The matrix $A\in \ensuremath{\mathbb{R}}^{10\times 10}$ is already defined in Maple. Find a basis of the image of $A$ as you normally would by hand (but use Maple for the computations!).
   2. Explore the Maple command ColumnSpace of the LinearAlgebra package: this command produces a basis of the image of the given matrix. Redo the previous item using this command.
   3. Find a basis of the kernel of $A$ as you normally would by hand (but use Maple for the computations!).
   4. Explore the Maple command NullSpace of the LinearAlgebra package: this command produces a basis of the kernel of the given matrix. Redo the previous item using this command.
   5. Check the formula $\dim \ker(M) + \dim {\mathrm{Im}}(M) = n$ for all $M\in \ensuremath{\mathbb{R}}^{n\times m}$ for the given $A$.

4. $B\in\ensuremath{\mathbb{R}}^{4\times 4}$ is already defined in Maple (as BB). Let

   $$U = \{ M\in \ensuremath{\mathbb{R}}^{4\times 4} : M\cdot B = B\cdot M\}.$$

   
   
   Find a basis of $U$. What is $\dim U$?