91Ó°ÊÓ

determine whether the set, together with the standard operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails. The set of all \(2 \times 2\) nonsingular matrices

In Exercises \(27-40\) find the nullspace of the matrix. \(A=\left[\begin{array}{rr}5 & 2 \\ 3 & -1 \\ 2 & 1\end{array}\right]\)

Determine whether the subset of \(M_{n, n}\) is a subspace of \(M_{n, n}\) with the standard operations. Justify your answer. The set of all \(n \times n\) matrices whose trace is nonzero (Recall that the trace of a matrix is the sum of the main diagonal entries of the matrix.)

determine whether the set, together with the standard operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails. \(C[0,1],\) the set of all continuous functions defined on the interval \([0,1]\)

(a) find the transition matrix from \(B\) to \(B^{\prime},(b)\) find the transition matrix from \(B^{\prime}\) to \(B\) (c) verify that the two transition matrices are inverses of each other, and (d) find the coordinate matrix \([\mathrm{x}]_{n},\) given the coordinate matrix \([\mathbf{x}]_{B^{* *}}\) $$\begin{aligned}&B=\\{(1,3),(-2,-2)\\}, B^{\prime}=\\{(-12,0),(-4,4)\\}\\\&[\mathbf{x}]_{E^{\prime}}=\left[\begin{array}{r}-1 \\\3\end{array}\right]\end{aligned}$$

Pendulum Consider a pendulum of length \(L\) that swings by the force of gravity only. For small values of \(\theta=\theta(t),\) the motion of the pendulum can be approximated by the differential equation $$ \frac{d^{2} \theta}{d t^{2}}+\frac{g}{L} \theta=0 $$ where \(g\) is the acceleration due to gravity. (a) Verify that $$ \\{\sin \sqrt{\frac{g}{L}} t, \cos \sqrt{\frac{g}{L}} t\\} $$ is a set of linearly independent solutions of the differential equation. (b) Find the general solution of the differential equation and show that it can be written in the form $$ \theta(t)=A \cos [\sqrt{\frac{g}{L}}(t+\phi)] $$

(a) find the transition matrix from \(B\) to \(B^{\prime},(b)\) find the transition matrix from \(B^{\prime}\) to \(B\) (c) verify that the two transition matrices are inverses of each other, and (d) find the coordinate matrix \([\mathrm{x}]_{n},\) given the coordinate matrix \([\mathbf{x}]_{B^{* *}}\) $$\begin{aligned}&B=\\{(1,1,1),(1,-1,1),(0,0,1)\\}\\\&B^{\prime}=\\{(2,2,0),(0,1,1),(1,0,1)\\}\\\&[\mathbf{x}]_{B^{r}}=\left[\begin{array}{l}2 \\\3 \\\1\end{array}\right]\end{aligned}$$

Proof Let \(\left\\{y_{1}, y_{2}\right\\}\) be a set of solutions of a second- order linear homogeneous differential equation. Prove that this set is linearly independent if and only if the Wronskian is not identically equal to zero.

Proof Prove in full detail that \(M_{2,2},\) with the standard operations, is a vector space. Rather than use the standard definitions of addition and scalar multiplication in \(R^{2}\), let these two operations be defined as shown below. A. \(\left(x_{1}, y_{1}\right)+\left(x_{2}, y_{2}\right)=\left(x_{1}+x_{2}, y_{1}+y_{2}\right)\) \(c(x, y)=(c x, y)\) B. \(\left(x_{1}, y_{1}\right)+\left(x_{2}, y_{2}\right)=\left(x_{1}, 0\right)\) \(c(x, y)=(c x, c y)\) C. \(\left(x_{1}, y_{1}\right)+\left(x_{2}, y_{2}\right)=\left(x_{1}+x_{2}, y_{1}+y_{2}\right)\) \(c(x, y)=(\sqrt{c} x, \sqrt{c} y)\) With each of these new definitions, is \(R^{2}\) a vector space? Justify your answers.

Writing Is the sum of two solutions of a nonhomogeneous linear differential equation also a solution? Explain.