Simplifying Linear Algebra: Techniques, Solutions, and Examples

This has four parts to it. the following system of linear equations
x - y =2z=13
2x +2y -z =-6
-x + 3y +z = -7
a) Provide a coefficient matrix corresponding to the system oflinear equations
b) what is the inverse of this matrix?
c) What is the transpose of the matrix?
d) find the determinant for this matrix?

If the position vectors of points T and S are 3a, - 2a, +a, and 4a, + 6a, + 2a, respectively, find (a) the coordinates of T and S, (b) the distance vector from T to S, (c) the distance between T and S.

Find the inverse of the following matrix A, if possible. Check that $A\ast <!--\; \ast \; -->A^{-<!--\; -\; -->1}$ and $A^{-<!--\; -\; -->1}\ast <!--\; \ast \; -->A=I$
$A=\left[\begin{array}{cc}4& 8\\ -<!--\; -\; -->5& -<!--\; -\; -->10\end{array}\right]$
The inverse, $A^{-<!--\; -\; -->1}$, is A=?

Let P be the plane containing the point(-5,-5,5) and with normal vector <3,2,1>. Find thepoint of P closest to the point (6,1,2).

Find the distance between the skew lines
$r_1(t)=(1;1;0)+(1,6,2)t$ and $r_2(s)=(1,5,2)+(2,15,6)s$.

Find a unit vector parallel to the vector y=<1,-2,2> which points in the opposite direction.

Find the unit vector along the line joining point (2, 4, 4) to point (-3,2,2).

Find the production matrix for the following input-output and demand matrices using the open model.
$A=\left[\begin{array}{cc}-<!--\; -\; -->0.1& 0.2\\ 0.55& 0.4\end{array}\right]$
$D=\left[\begin{array}{c}3\\ 4\end{array}\right]$

Solve the following system of equations by using the inverse of the coefficient matrix A.
(AX=B), x+5y=-10, -2x+7y=-31

If A is an n x n matrix , where are the entries on the main diagonal of A-A^T? Justify yoyr answer.

Let $T:M_{2x2}\to <!--\; \backslash rightarrow\; -->{\mathbb{R}}^3$ have matrix $[T{]}_{B,A}=\left[\begin{array}{cccc}1& 2& 0& 1\\ 0& 1& 1& 0\\ 1& 1& -<!--\; -\; -->1& -<!--\; -\; -->1\end{array}\right]$ relative to $A=\{\left[\begin{array}{cc}2& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 3\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 5& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 6\end{array}\right]\}$ and $\mathrm{\beta <!--\; \beta \; -->}=\{(1,1,1),(1,2,3),(1,4,9)\}$. Find the matrix of T relative to the bases $A^{\prime }=\{\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 4\\ 0& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 2& 0\end{array}\right],\left[\begin{array}{cc}0& 0\\ 0& 7\end{array}\right]\}$ and ${\mathrm{\beta <!--\; \beta \; -->}}^{\prime }=\{(1,1,1),(1,0,0),(1,1,0)\}$

representative matrix of a linear transformation
given a linear transformation: $T:M_n(\mathbb{C})\to <!--\; \to \; -->M_n(\mathbb{C})$, $T(A)=A-<!--\; -\; -->2A^T$, what is the

Let V be a 3 dimensional vector space over a field F and fix $({\mathbf{v}}_{\mathbf{1}},{\mathbf{v}}_{\mathbf{2}},{\mathbf{v}}_{\mathbf{3}})$ as a basis. Consider a linear transformation $T:V\to <!--\; \backslash rightarrow\; -->V$. Then we have
$T({\mathbf{v}}_{\mathbf{1}})=a_{11}{\mathbf{v}}_{\mathbf{1}}+a_{21}{\mathbf{v}}_{\mathbf{2}}+a_{31}{\mathbf{v}}_{\mathbf{3}}$
$T({\mathbf{v}}_{\mathbf{2}})=a_{12}{\mathbf{v}}_{\mathbf{1}}+a_{22}{\mathbf{v}}_{\mathbf{2}}+a_{32}{\mathbf{v}}_{\mathbf{3}}$
$T({\mathbf{v}}_{\mathbf{3}})=a_{13}{\mathbf{v}}_{\mathbf{1}}+a_{23}{\mathbf{v}}_{\mathbf{2}}+a_{33}{\mathbf{v}}_{\mathbf{3}}$
So that we can identify T by the matrix
$\left(\begin{array}{ccc}{a}_{11}& a_{12}& a_{13}\\ a_{21}& a_{22}& a_{23}\\ a_{31}& a_{32}& a_{33}\end{array}\right)$
But then when I read several linear algebra book, it said if $T({\mathbf{v}}_{\mathbf{i}})=\underset{j}{\sum <!--\; \sum \; -->}{a}_{ij}{\mathbf{v}}_{\mathbf{j}}$ , then we can identify T by the matrix $(a_{ij})$. My problem is: isn't the matrix is $(a_{ji})$ instead of $(a_{ij})$?

Let A be the matrix below and define a transformation $T:{\mathbb{R}}^3\to <!--\; \backslash rightarrow\; -->{\mathbb{R}}^3$ by $T(U)=AU.$. For each of the vectors $B$ below, find a vector $U$ such that $T$ maps $U$ to $B$, if possible. Otherwise state that there is no such $U$.
$\left(\begin{array}{ccc}1& -<!--\; -\; -->3& 2\\ 2& -<!--\; -\; -->4& 4\\ 3& -<!--\; -\; -->8& 6\end{array}\right)=A$
a) $\left(\begin{array}{c}4\\ 6\\ 11\end{array}\right)=B$
b) $\left(\begin{array}{c}-<!--\; -\; -->3\\ -<!--\; -\; -->2\\ -<!--\; -\; -->7\end{array}\right)=B$

Suppose $T$ is a transformation from ${\mathbb{R}}^2$ to ${\mathbb{R}}^2$. Find the matrix $A$ that induces $T$ if $T$ is the (counter-clockwise) rotation by $\frac{3}{4}\mathrm{\pi <!--\; \pi \; -->}$.
how to begin to find a matrix that is 2x2 for this question.

Let $A$ be the set of all $n\times <!--\; \times \; -->n$ symmetric real matirix and $f\in <!--\; \in \; -->C(\mathbb{R},A)$. Then whether there is a $g\in <!--\; \in \; -->C(\mathbb{R},O(n))$ such that for all $t\in <!--\; \in \; -->\mathbb{R}$, $g(t{)}^{-<!--\; -\; -->1}f(t)g(t)$ is a diagonal matrix?

Can't figure out this transformation matrix
1) The positive z axis normalized as Vector(0,0,1) has to map to an arbitrary direction vector in the new coordinate system Vector(a,b,c)
2) The origin in the original coordinate system has to map to an arbitrary position P in the new coordinate system.
3) This might be redundant but the positive Y axis has to map to a specific direction vector(d,e,f) which is perpendicular to Vector(a,b,c) from before.
So my question is twofold: 1) How would I go about constructing this transformation matrix and 2) Is this enough data to ensure that any arbitrary vector in coordinate system 1 will be accurately transformed in coordinate system 2?