Affine transformations

MathProblemsBank banner

MathProblemsBank banner

Math Problems and solutions

Problem list Free problems

Attention! If a subsection is selected, then the search will be performed in it!

Problem: Write a parametric and algebraic affine span of sets. \[ \{(1,1,1,1,1),(2,2,2,2,2),(1,-1,2,-2,3)\} \text {. } \]

5.1.1 Affine transformations

3.05 $

Problem: The plane \( P: x_{1}+2 x_{2}+3 x_{3}+a x_{4}=5 \) is parallel to the plane \( Q: x_{1}+2 x_{2}=1, x_{3}+x_{4}=2 \) for \( a=\cdots \).

5.1.2 Affine transformations

3.82 $

Problem: Affine span of a set \( \left\{(1,1,1,1),\left(1, \frac{3}{2}, 1, \frac{3}{2}\right),(2,1,2,1)\right\} \) parallel to the line passing through the points \( (1,0,0,0) \) and \( (3, a, 2,1) \), for \( a=\cdots \).

5.1.3 Affine transformations

3.05 $

Problem: The improper motion is given by \[ X^{\prime}=\frac{1}{3}\left(\begin{array}{ccc} 2 & 2 & -1 \\ 2 & -1 & 2 \\ -1 & 2 & 2 \end{array}\right) X+\left(\begin{array}{l} 4 \\ 0 \\ 2 \end{array}\right) \] It is a composition of symmetry with respect to the plane \( 2 x_{1}-4 x_{2}+a x_{3}=6 \) and parallel translation, moreover \( a=\cdots \).

5.1.4 Affine transformations

3.82 $

Problem: To obtain the canonical form of the hypersurface given by the equation \( 2 x_{1} x_{2}+x_{2}-x_{3}=0 \), it's necessary to move the origin of the coordinates to the point \( (a, b, c) \), where \( 2(a+b+c)=\cdots \).

5.1.5 Affine transformations

2.54 $

Problem: To obtain the canonical form of the hypersurface given by the equation \( 2 x_{1} x_{2}+2 x_{2}+x_{3}+1=0 \), it's necessary to move the origin of coordinates to the point \( (a, b, c) \), where \( a++b+c=\cdots \).

5.1.6 Affine transformations

2.54 $

Problem: Find the square of the distance from the point \( A(1,0,1,3) \) to the plane \( P: x_{1}+x_{2}+x_{3}+2 x_{4}=7 \).

5.1.7 Affine transformations

1.27 $

Problem: Find the square of the distance from the point \( A= \) \( (1,0,0,1) \) to the plane given by the system of equations \[ x_{1}-x_{2}-x_{3}+x_{4}=0,2 x_{1}+x_{2}-x_{3}+x_{4}=1 \text {. } \]

5.1.8 Affine transformations

3.05 $

Problem: Find the square of the distance from the point \( A= \) \( (2,0,1,2) \) to the plane \[ P:\left(\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{array}\right)=\left(\begin{array}{l} 1 \\ 1 \\ 2 \\ 2 \end{array}\right)+t_{1}\left(\begin{array}{l} 1 \\ 0 \\ 0 \\ 1 \end{array}\right)+t_{2}\left(\begin{array}{l} 0 \\ 1 \\ 0 \\ 1 \end{array}\right)+t_{3}\left(\begin{array}{l} 0 \\ 0 \\ 1 \\ 1 \end{array}\right) . \]

5.1.9 Affine transformations

3.05 $