College Algebra Solutions for Your Success

Show that an element and its inverse have the same order in any group.

The coordinates of the point in the $x^{\text{'}} y^{\text{'}}$ - coordinate system with the given angle of rotation and the xy-coordinates.

Consider the bases $B=\left(\begin{array}{c}\left[\begin{array}{c}2\\ 3\end{array}\right],\left[\begin{array}{c}3\\ 5\end{array}\right]\end{array}\right)of{R}^{2}andC=(\begin{array}{c}\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right]\end{array},\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right])of{R}^3$.
and the linear maps $S\in L(R^2,R^3)andT\in L(R^3,R^2)$ given given (with respect to the standard bases) by $[S{]}_{E,E}=\left[\begin{array}{cc}2& -1\\ 5& -3\\ -3& 2\end{array}\right]and[T{]}_{E,E}=\left[\begin{array}{ccc}1& -1& 1\\ 1& 1& -1\end{array}\right]$ Find each of the following coordinate representations. ${\displaystyle \left(b\right){\left[S\right]}_{E,C}}$
${\displaystyle \left(c\right){\left[S\right]}_{B,C}}$

The given system of inequality:
${\displaystyle \{\begin{array}{c}y<9-x^2\\ y\ge x+3\end{array}}$
Also find the coordinates of all vertices, and check whether the solution set is bounded.

Consider the following linear transformation $T:P_2\to P_3$, given by $T(f)=3x^{2}f$'.

That is, take the first derivative and then multiply by $3x^2$

(a) Find the matrix for T with respect to the standard bases of $P_n$: that is, find $[T{]}_{ϵ}^{ϵ}$, where- $ϵ=1,x,x^2,x^n$

(b) Find N(T) and R(T). You can either work with polynomials or with their coordinate vectors with respect to the standard basis. Write the answers as spans of polynomials.

(c) Find the the matrix for T with respect to the alternate bases: $[T{]}_A^B$ where $A=x-1,x,x^2+1,B=x^3,x,x^2,1.$