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Question 1: Solving Systems of Linear Equations

1(a) Generate a random matrix $A$ and vector $b$, find the RREF and the solution set.

We generate a random 5x6 matrix $A$ and a random 5x1 vector $b$. The augmented matrix $[A|b]$ is then computed and its Reduced Row Echelon Form (RREF) is determined using MATLAB to find the solution set.

1(b) Answer textbook question Section 2.1 B44.

Question 2: Linear/Matrix Mappings and Projections

2(a) Project vectors onto $\mathbf{v}=(1,2,3)$.

The projection matrix $P$ is given by: $P=\frac{\mathbf{v}{\mathbf{v}}^T}{|\mathbf{v}{|}^2}$ We compute the projections of ${\mathbf{x}}_1=(3,4,1)$, ${\mathbf{x}}_2=(2,4,6)$, and ${\mathbf{x}}_3=(1,1,-1)$ using MATLAB.

2(b) Composition of rotation and projection matrices.

Given the rotation matrix $R$: $R=\left[\begin{array}{ccccccc}0.8& 0& 0.60& 1& 0-0.6& 0& 0.8\end{array}\right]$ We calculate the composition matrices $RP$ (projection followed by rotation) and $PR$ (rotation followed by projection) using MATLAB.

Question 3: Basis from Column Space

3(a) Find the basis for the subspace $S=\text{span}{\mathbf{v}}_1,{\mathbf{v}}_2,{\mathbf{v}}_3,{\mathbf{v}}_4,{\mathbf{v}}_5$.

Given the vectors: ${\mathbf{v}}_1=\left[\begin{array}{c}3\\ 2\\ 0\\ -1\end{array}\right],{\mathbf{v}}_2=\left[\begin{array}{c}6\\ 5\\ 2\\ 1\end{array}\right],{\mathbf{v}}_3=\left[\begin{array}{c}3\\ 0\\ -4\\ -7\end{array}\right],{\mathbf{v}}_4=\left[\begin{array}{c}2\\ 0\\ -1\\ -1\end{array}\right],{\mathbf{v}}_5=\left[\begin{array}{c}-2\\ 9\\ 9\\ 6\end{array}\right]$ We construct matrix $A$ using these vectors as columns and find its RREF to identify the basis vectors for subspace $S$.

3(b) Express non-basis vectors as a linear combination.

Using the RREF results from part (3a), we identify the non-basis vectors and express them as a linear combination of the basis vectors identified from the pivot columns in the RREF.