Change of basis and coordinate system

In the plane, two bases $\mathbf{e}_{1}, \mathbf{e}_{2}$ and $\mathbf{e}_{1}^{\prime }, \mathbf{e}_{2}^{\prime }$ are given. The vectors of the second basis have coordinates $(- 1, 3)$ and $(2, - 7)$ respectively in the first basis.

In space, two bases $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ and $\mathbf{e}_{1}^{\prime }, \mathbf{e}_{2}^{\prime }, \mathbf{e}_{3}^{\prime }$ are given. The vectors of the second basis have coordinates $(1, 1, 1), (- 1, - 2, - 3), (1, 3, 6)$ respectively in the first basis.

In the plane, two coordinate systems $O$, $\mathbf{e}_{1}, \mathbf{e}_{2}$ and $O^{\prime }, \mathbf{e}_{1}^{\prime }, \mathbf{e}_{2}^{\prime }$ are given. The origin of the second coordinate system has coordinates $(- 1, 3)$ in the first system, and the basis vectors of the second system have coordinates $(2, 3)$ and $(1, 1)$ respectively in the basis of the first system.

In space, two coordinate systems $O, \mathbf{e}_{1}, \mathbf{e}_{2}$, $\mathbf{e}_{3}$ and $O^{\prime }, \mathbf{e}_{1}^{\prime }, \mathbf{e}_{2}^{\prime }, \mathbf{e}_{3}^{\prime }$ are given. The origin of the second coordinate system has coordinates $(1, 1, 2)$ in the first system, and the basis vectors of the second coordinate system have coordinates $(4, 2, 1), (5, 3, 2), (3, 2, 1)$ respectively in the basis of the first system.

The coordinates $x, y$ of every point of the plane in the coordinate system $O, \mathbf{e}_{1}, \mathbf{e}_{2}$ are expressed through the coordinates $x^{\prime }, y^{\prime }$ of the same point in the system $O^{\prime }, \mathbf{e}_{1}^{\prime }, \mathbf{e}_{2}^{\prime }$ by the formulas $x = 2x^{\prime } - y^{\prime } + 5$, $y = 3x^{\prime } + y^{\prime } + 2$.

The coordinates $x, y, z$ of every point of space in the coordinate system $O, \mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ are expressed through the coordinates $x^{\prime }, y^{\prime }, z^{\prime }$ of the same point in the system $O^{\prime }, \mathbf{e}_{1}^{\prime }, \mathbf{e}_{2}^{\prime }, \mathbf{e}_{3}^{\prime }$ by the formulas $x = x^{\prime } + y^{\prime } + z^{\prime } - 1, y = - x^{\prime } + z^{\prime } + 3, z = - x^{\prime } - y^{\prime } - 2$.

Find the coordinates of a vector in the basis $\mathbf{e}_{1}(2, 3), \mathbf{e}_{2}(3, 4)$ in the plane, if its coordinates $\alpha_{1}^{\prime }, \alpha_{2}^{\prime }$ in the basis $\mathbf{e}_{1}^{\prime }(1, - 1), \mathbf{e}_{2}^{\prime }(2, - 3)$ are known.

Find the coordinates of a vector in the basis $\mathbf{e}_{1}(1, 3, 2)$, $\mathbf{e}_{2}(- 1, 1, 0), \mathbf{e}_{3}(2, - 1, 1)$ in space, if its coordinates $\alpha_{1}^{\prime }, \alpha_{2}^{\prime }, \alpha_{3}^{\prime }$ in the basis $\mathbf{e}_{1}^{\prime }(- 1, 0, 2), \mathbf{e}_{2}^{\prime }(1, 1, 1), \mathbf{e}_{3}^{\prime }(4, 3, - 1)$ are known.

Find the coordinates of a point in the coordinate system $O(2, - 1)$, $\mathbf{e}_{1}(1, 5), \mathbf{e}_{2}(- 1, 4)$ in the plane, if its coordinates $x^{\prime }, y^{\prime }$ in the coordinate system $O^{\prime }(3, 2), \mathbf{e}_{1}^{\prime }(1, - 1), \mathbf{e}_{2}^{\prime }(4, 2)$ are known.

Find the coordinates of a point in the coordinate system $O(1, 3$, $3), \mathbf{e}_{1}(3, 3, 1), \mathbf{e}_{2}(3, 5, 2), \mathbf{e}_{3}(1, 2, 1)$ in space, if its coordinates $x^{\prime }, y^{\prime }, z^{\prime }$ in the coordinate system $O^{\prime }(- 1, 0, 2)$, $\mathbf{e}_{1}^{\prime }(1, - 2, 1), \mathbf{e}_{2}^{\prime }(4, 2, 1), \mathbf{e}_{3}^{\prime }(2, - 1, 3)$ are known.

In parallelogram $ABCD$, point $E$ lies on diagonal $BD$, with $\left|BE\right|:\left|ED\right| = 1: 2$. Find the coordinates of a point of the plane in the coordinate system $A, \overrightarrow {AB}, \overrightarrow {AD}$, if its coordinates $x^{\prime }, y^{\prime }$ in the coordinate system $E, \overrightarrow {EC}, \overrightarrow {ED}$ are known.

In parallelogram $ABCD$, point $E$ lies on side $BC$, and point $F$ lies on side $AB$, with $\left|BE\right|:\left|BC\right| = 1: 4, \left|BF\right|:\left|AF\right| = 2: 5$. Find the coordinates of a point of the plane in the coordinate system $C, \overrightarrow {CE}, \overrightarrow {CD}$, if its coordinates $x^{\prime }, y^{\prime }$ in the coordinate system $E, \overrightarrow {EF}, \overrightarrow {ED}$ are known.

In triangle $ABC$, point $D$ lies on side $BC$, and point $E$ lies on the extension of side $AC$ beyond point $C$, with $\left|BD\right|:\left|DC\right| = 1: 2, \left|AC\right|:\left|CE\right| = 3: 1$. Find the coordinates of a point of the plane in the coordinate system $A, \overrightarrow {AB}, \overrightarrow {AC}$, if its coordinates $x^{\prime }, y^{\prime }$ in the coordinate system $D, \overrightarrow {DA}, \overrightarrow {DE}$ are known.

In triangle $ABC$, point $D$ lies on side $AC$, and point $E$ lies on segment $BD$, with $\left|AD\right|:\left|AC\right| = 1: 3$, $\left|BE\right|:\left|ED\right| = 2: 3$. Find the coordinates of a point of the plane in the coordinate system $A, \overrightarrow {AB}, \overrightarrow {AD}$, if its coordinates $x^{\prime }$, $y^{\prime }$ in the coordinate system $C, \overrightarrow {CB}, \overrightarrow {CE}$ are known.

Given a regular hexagon $ABCDEF$. Find the coordinates of a point of the plane in the coordinate system $A, \overrightarrow {AB}, \overrightarrow {AF}$, if its coordinates $x^{\prime }, y^{\prime }$ in the coordinate system $C$, $\overrightarrow {CB}, \overrightarrow {CE}$ are known.

In trapezoid $ABCD$ the diagonals meet at point $E$, and the lengths of the bases $BC$ and $AD$ are in the ratio $2: 3$. Find the coordinates of a point of the plane in the coordinate system $A, \overrightarrow {AB}$, $\overrightarrow {AD}$, if its coordinates $x^{\prime }, y^{\prime }$ in the coordinate system $E$, $\overrightarrow {EA}, \overrightarrow {EB}$ are known.

In trapezoid $ABCD$ the lengths of the bases $BC$ and $AD$ are in the ratio $3: 4$, point $E$ is the midpoint of base $AD$, and the extensions of the lateral sides meet at point $F$. Find the coordinates of a point of the plane in the coordinate system $E, \overrightarrow {EB}$, $\overrightarrow {EC}$, if its coordinates $x^{\prime }, y^{\prime }$ in the coordinate system $F, \overrightarrow {FB}, \overrightarrow {FC}$ are known.

The base of prism $ABCDA_{1} B_{1} C_{1} D_{1}$ is a rhombus with acute angle $A$ equal to $60^{\circ }$. Point $K$ lies on the extension of edge $AB$ beyond point $B$, with angle $ADK$ right. Find the coordinates of a point of space in the coordinate system $A, \overrightarrow {AB}, \overrightarrow {AD}$, $\overrightarrow {AA_{1}}$, if its coordinates $x^{\prime }, y^{\prime }, z^{\prime }$ in the coordinate system $K, \overrightarrow {KA}, \overrightarrow {KD}, \overrightarrow {KC_{1}}$ are known.

In triangular prism $ABCA_{1} B_{1} C_{1}$, point $M$ is the intersection point of the medians of face $A_{1} B_{1} C_{1}$. Find the coordinates of a point of space in the coordinate system $A, \overrightarrow {AB}, \overrightarrow {AC}, \overrightarrow {AB_{1}}$, if its coordinates $x^{\prime }, y^{\prime }, z^{\prime }$ in the coordinate system $A_{1}, \overline{A_{1} B}$, $\overrightarrow {A_{1} C}, \overrightarrow {A_{1} M}$ are known.

In tetrahedron $ABCD$, point $M$ is the intersection point of the medians of face $BCD$. Find the coordinates of a point of space in the coordinate system $A, \overrightarrow {AB}, \overrightarrow {AC}, \overrightarrow {AD}$, if its coordinates $x^{\prime }, y^{\prime }, z^{\prime }$ in the coordinate system $M, \overrightarrow {MB}, \overrightarrow {MC}, \overrightarrow {MA}$ are known.

In a regular hexagonal pyramid $SABCDEF$ with apex $S$, point $M$ is the center of the base. Find the coordinates of a point of space in the coordinate system $A, \overrightarrow {AB}, \overrightarrow {AF}$, $\overrightarrow {AS}$, if its coordinates $x^{\prime }, y^{\prime }, z^{\prime }$ in the coordinate system $S, \overrightarrow {SC}, \overrightarrow {SD}, \overrightarrow {SM}$ are known.

Given a parallelepiped $ABCDA_{1} B_{1} C_{1} D_{1}$. Find the coordinates of a point of space in the coordinate system $A, \overrightarrow {AC}, \overrightarrow {AB_{1}}$, $\overrightarrow {AA_{1}}$, if its coordinates $x^{\prime }, y^{\prime }, z^{\prime }$ in the coordinate system $D_{1}, \overrightarrow {D_{1} D}, \overrightarrow {D_{1} C_{1}}, \overrightarrow {D_{1} \bar{B}}$ are known

The coordinates $x, y$ of every point of the plane in the first coordinate system are expressed through the coordinates $x^{\prime }, y^{\prime }$ of the same point in the second coordinate system by the relations $x = a_{11} x^{\prime } + a_{12} y^{\prime } + a_{10}, y = a_{21} x^{\prime } + a_{22} y^{\prime } + a_{20}$. The first coordinate system is rectangular. Under what necessary and sufficient condition is the second system also rectangular?

The coordinates $x, y, z$ of every point of space in the first coordinate system are expressed through the coordinates $x^{\prime }$, $y^{\prime }, z^{\prime }$ of the same point in the second coordinate system by the relations

In the plane, two rectangular coordinate systems $O, \mathbf{e}_{1}, \mathbf{e}_{2}$ and $O^{\prime }, \mathbf{e}_{1}^{\prime }, \mathbf{e}_{2}^{\prime }$ are given. The origin of the second coordinate system has coordinates $x_{0}, y_{0}$ in the first system, and the vectors $\mathbf{e}_{1}^{\prime }$ and $\mathbf{e}_{2}^{\prime }$ are obtained from the vectors $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$ respectively by a rotation through the same angle $\varphi$ in the direction of the shortest rotation from $\mathbf{e}_{1}$ to $\mathbf{e}_{2}$.

In the plane, two rectangular coordinate systems $O, \mathbf{e}_{1}, \mathbf{e}_{2}$ and $O^{\prime }, \mathbf{e}_{1}^{\prime }, \mathbf{e}_{2}^{\prime }$ are given. The origin of the second coordinate system has coordinates 1, 3 in the first system, and the vectors $\mathbf{e}_{1}^{\prime }$ and $\mathbf{e}_{2}^{\prime }$ are obtained from the vectors $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$ respectively by a rotation through the same angle $\varphi$ in the direction of the shortest rotation from $\mathbf{e}_{1}$ to $\mathbf{e}_{2}$. Find the coordinates of a point in the first coordinate system, if its coordinates $x^{\prime }, y^{\prime }$ in the second system are known, taking the angle $\varphi$ equal to:

In the plane, two rectangular coordinate systems $O, \mathbf{e}_{1}, \mathbf{e}_{2}$ and $O^{\prime }, \mathbf{e}_{1}^{\prime }, \mathbf{e}_{2}^{\prime }$ are given. The origin of the second coordinate system has coordinates $x_{0}, y_{0}$ in the first system, and the vectors $\mathbf{e}_{1}^{\prime }$ and $- \mathbf{e}_{2}^{\prime }$ are obtained from the vectors $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$ respectively by a rotation through the same angle $\varphi$ in the direction of the shortest rotation from $\mathbf{e}_{1}$ to $\mathbf{e}_{2}$.

In right triangle $ABC$, whose legs have lengths $\left|AB\right| = 3$ and $\left|BC\right| = 4$, point $D$ is the foot of the altitude drawn from the vertex of the right angle. Vectors $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{1}^{\prime }, \mathbf{e}_{2}^{\prime }$ have length 1, with $\mathbf{e}_{1}$ equally directed with $\overrightarrow {BA}, \mathbf{e}_{2}$ equally directed with $\overrightarrow {BC}, \mathbf{e}_{1}^{\prime }$ equally directed with $\overrightarrow {AC}, \mathbf{e}_{2}^{\prime }$ equally directed with $\overrightarrow {DB}$. Find the coordinates of a point of the plane in the coordinate system $B, \mathbf{e}_{1}$, $\mathbf{e}_{2}$, if its coordinates $x^{\prime }, y^{\prime }$ in the coordinate system $D$, $\mathbf{e}_{1}^{\prime }, \mathbf{e}_{2}^{\prime }$ are known.

In space, two rectangular coordinate systems $O, \mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ and $O^{\prime }, \mathbf{e}_{1}^{\prime }, \mathbf{e}_{2}^{\prime }, \mathbf{e}_{3}^{\prime }$ are given. The origin of the second coordinate system has coordinates $- 1, 3, 5$ in the first system. The vector $\mathbf{e}_{1}^{\prime }$ makes angles equal to $60^{\circ }$ with the vectors $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$, and an acute angle with the vector $\mathbf{e}_{3}$. The vector $\mathbf{e}_{2}^{\prime }$ is coplanar with the vectors $\mathbf{e}_{1}$ and $\mathbf{e}_{2}$ and makes an acute angle with the vector $\mathbf{e}_{2}$. The triples $\mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ and $\mathbf{e}_{1}^{\prime }, \mathbf{e}_{2}^{\prime }, \mathbf{e}_{3}^{\prime }$ have the same orientation. Find the coordinates of a point of space in the first coordinate system, if its coordinates $x^{\prime }, y^{\prime }, z^{\prime }$ in the second system are known.

In space, two rectangular coordinate systems $O, \mathbf{e}_{1}, \mathbf{e}_{2}, \mathbf{e}_{3}$ and $O^{\prime }, \mathbf{e}_{1}^{\prime }, \mathbf{e}_{2}^{\prime }, \mathbf{e}_{3}^{\prime }$ are given. Points $O$ and $O^{\prime }$ are distinct, and the endpoints of the vectors $\mathbf{e}_{2}$ and $\mathbf{e}_{i}^{\prime }$, drawn respectively from points $O$ and $O^{\prime }$, coincide $(i = 1, 2, 3)$. Find the coordinates of a point of space in the first coordinate system, if its coordinates $x^{\prime }, y^{\prime }, z^{\prime }$ in the second system are known.