LINEAR ALGEBRA (Math 211)
HW #12 (Due Wed. Nov. 8 )


1. Find the dimension of the linear space of all 2 x 2 matrices and give an example of a basis in it.


2. Prove that the vectors $\mbox{\bf v}_1=(1,1,1)$, $\mbox{\bf v}_2=(1,1,2)$, and $\mbox{\bf v}_3=(1,2,3)$ form a basis in ${\mathbb R}^3$ and compute the coordinates of vector $\mbox{\bf x}=(6,9,14)$ with respect to this basis.


3. For vectors $\mbox{\bf v}_1=(1,2,0,4)$, $\mbox{\bf v}_2=(-1,0,5,1)$, and $\mbox{\bf v}_3=(1,6,10,14)$ find the dimension $\dim\left(\mbox{Span}(\mbox{\bf v}_1, \mbox{\bf v}_2, \mbox{\bf v}_3)\right)$. Which of the vectors $\mbox{\bf v}_1$, $\mbox{\bf v}_2$, $\mbox{\bf v}_3$ form a basis of this space?


4. Find the rank of matrix $\left(\begin{array}{rrrr} 1&2&3&4\\ 2&3&4&5\\ 3&4&5&6\\ 4&5&6&7 \end{array}\right)$.


5. Find a basis of the kernel of matrix $\left(\begin{array}{rrrrr} 3&4&1&2&3\\ 5&7&1&3&4\\ 4&5&2&1&5\\ 7&10&1&6&5 \end{array}\right)$.


6. Let $\mbox{\bf0}$ be the zero vector. Prove that the vectors $\mbox{\bf0}$, $\mbox{\bf v}_1$, $\mbox{\bf v}_2$, ..., $\mbox{\bf v}_r$ are always linearly dependent for any vectors $\mbox{\bf v}_1$, $\mbox{\bf v}_2$, ..., $\mbox{\bf v}_r$.