91Ó°ÊÓ

In each part, determine whether the given vector is in the span of \(S\). (a) \((2,-1,1), \quad S=\\{(1,0,2),(-1,1,1)\\}\) (b) \((-1,2,1), \quad S=\\{(1,0,2),(-1,1,1)\\}\) (c) \((-1,1,1,2), \quad S=\\{(1,0,1,-1),(0,1,1,1)\\}\) (d) \((2,-1,1,-3), \quad S=\\{(1,0,1,-1),(0,1,1,1)\\}\) (e) \(-x^{3}+2 x^{2}+3 x+3, \quad S=\left\\{x^{3}+x^{2}+x+1, x^{2}+x+1, x+1\right\\}\) (f) \(2 x^{3}-x^{2}+x+3, \quad S=\left\\{x^{3}+x^{2}+x+1, x^{2}+x+1, x+1\right\\}\) (g) \(\left(\begin{array}{rr}1 & 2 \\ -3 & 4\end{array}\right), \quad S=\left\\{\left(\begin{array}{rr}1 & 0 \\ -1 & 0\end{array}\right),\left(\begin{array}{ll}0 & 1 \\ 0 & 1\end{array}\right),\left(\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right)\right\\}\) (h) \(\left(\begin{array}{ll}1 & 0 \\ 0 & 1\end{array}\right), \quad S=\left\\{\left(\begin{array}{rr}1 & 0 \\ -1 & 0\end{array}\right),\left(\begin{array}{ll}0 & 1 \\ 0 & 1\end{array}\right),\left(\begin{array}{ll}1 & 1 \\ 0 & 0\end{array}\right)\right\\}\)

Prove that \(A+A^{t}\) is symmetric for any square matrix \(A\).

Exercises 5 and 6 show why the definitions of matrix addition and scalar multiplication (as defined in Example 2) are the appropriate ones. Richard Gard ("Effects of Beaver on Trout in Sagehen Creek, California," J. Wildlife Management, 25, 221-242) reports the following number of trout having crossed beaver dams in Sagehen Creek. Upstream Crossings $$ \begin{array}{lccc} \hline & \text { Fall } & \text { Spring } & \text { Summer } \\ \hline \text { Brook trout } & 8 & 3 & 1 \\ \text { Rainbow trout } & 3 & 0 & 0 \\ \text { Brown trout } & 3 & 0 & 0 \\ \hline \end{array} $$ Upstream Crossings $$ \begin{array}{lccc} \hline & \text { Fall } & \text { Spring } & \text { Summer } \\ \hline \text { Brook trout } & 9 & 1 & 4 \\ \text { Rainbow trout } & 3 & 0 & 0 \\ \text { Brown trout } & 1 & 1 & 0 \\ \hline \end{array} $$Record the upstream and downstream crossings in two 3 x 3 matrices, and verify that the sum of these matrices gives the total number of crossings (both upstream and downstream) categorized by trout species and season

Prove that if the vector \(x\) emanates from the origin of the Euclidean plane and terminates at the point with coordinates \(\left(a_{1}, a_{2}\right)\), then the vector \(t x\) that emanates from the origin terminates at the point with coordinates \(\left(t a_{1}, t a_{2}\right)\). Visit goo.gl/eYTxuU for a solution.

Is \(\\{(1,4,-6),(1,5,8),(2,1,1),(0,1,0)\\}\) a linearly independent subset of \(R^{3} ?\) Justify your answer.

Give three different bases for \(\mathrm{F}^{2}\) and for \(\mathrm{M}_{2 \times 2}(F)\).

Show that the midpoint of the line segment joining the points \((a, b)\) and \((c, d)\) is \(((a+c) / 2,(b+d) / 2)\).

Prove that the diagonals of a parallelogram bisect each other.

Prove the following generalization of the replacement theorem. Let \(\beta\) be a basis for a vector space \(\mathrm{V}\), and let \(S\) be a linearly independent subset of V. There exists a subset \(S_{1}\) of \(\beta\) such that \(S \cup S_{1}\) is a basis for \(\mathrm{V}\).

Let \(S=\\{0,1\\}\) and \(F=R\). In \(\mathcal{F}(S, R)\), show that \(f=g\) and \(f+g=h\), where \(f(t)=2 t+1, g(t)=1+4 t-2 t^{2}\), and \(h(t)=5^{t}+1 .\)