de-CH
utf-8
math math-format
Determine Matrix Mapping
la-01-01a
multiple
972
randRangeExclude(-5,5,[-1,0,1]) randRangeExclude(-5,5,[-1,0,1]) randRangeExclude(-5,5,[-1,0,1]) randRangeExclude(-5,5,[-1,0,1])
randRangeNonZero(-5,5) randRangeNonZero(-5,5)

Given the vector v.

For which entries does the matrix A = \begin{pmatrix} A & \color{red}{b} \\ \color{blue}{c} & D \end{pmatrix} fit the situation below, if the second vector \color{orange} A\cdot v is?



graphInit({ range: [[-8, 8],[-8, 8]], scale: [25,25], tickStep: [1,1], gridStep: [1,1], labelStep: [2,2], gridOpacity: 0.1, axisOpacity: 0.8, tickOpacity: 0.6, labelOpacity: 0.8 }); label( [ 0, 8 ], "y", "above" ); label( [8,0 ], "x", "right" ); line( [0, 0], [X*A+B*Y, X*C+D*Y], { stroke: ORANGE, arrows: "->" } ); line( [0,0], [X, Y], { stroke: BLACK, arrows: "->" } ); label( [X + 0.5, Y + 0.5], "{v}", "left");

b \color{red}b = B
c \color{blue}c = C

Find the entries \color{red}b and \color{blue}c with A \cdot \begin{pmatrix} X \\ Y \end{pmatrix} = \color{orange}\begin{pmatrix} X*A+B*Y \\ X*C+D*Y \end{pmatrix}.

Initially, this is the equation \begin{pmatrix} negParens(X) \cdot negParens(A) + negParens(Y) \cdot {\color{red}b} \\ negParens(X) \cdot {\color{blue}c} + negParens(Y) \cdot negParens(D) \end{pmatrix} = \color{orange}\begin{pmatrix} X*A+B*Y \\ X*C+D*Y \end{pmatrix}.

In the two coordinates, the two equations are X*A + negParens(Y) \cdot {\color{red}b} = \color{orange} X*A+B*Y and negParens(X) \cdot {\color{blue}c} + Y*D = \color{orange} X*C+D*Y.

Therefore \color{red}b =B and \color{blue}c =C.