$\overrightarrow{w}=M\cdot \overrightarrow{v}=\left(\begin{array}{cc}{m}_{11}& {m}_{12}\\ {m}_{21}& {m}_{22}\end{array}\right)\cdot \left(\begin{array}{c}{v}_{x}\\ {v}_{y}\end{array}\right)$

The operation with the vector is graphically presented. By moving the vector endpoint the vector can be changed. The sliders change the matrix elements. The red vector is the result vector of the multiplication with the matrix.

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m_{11} =

m_{12} =

v_{x} =

m_{21} =

m_{22} =

v_{y} =

The product of a matrix with a vector is a linear image. The multiplication is explained if the number of columns of the matrix is equal to the number of elements of the vector. The result is a vector whose number of components equals the number of rows of the matrix. This means that a matrix with 2 rows always maps a vector to a vector with two components.

$A\cdot \overrightarrow{v}=\left(\begin{array}{cccc}{a}_{11}& {a}_{12}& \dots & {a}_{1m}\\ {a}_{21}& {a}_{22}& \dots & {a}_{2m}\\ & \vdots \\ {a}_{n1}& {a}_{n2}& \dots & {a}_{nm}\end{array}\right)\cdot \left(\begin{array}{c}{v}_{1}\\ {v}_{2}\\ \vdots \\ {v}_{m}\end{array}\right)=\left(\begin{array}{c}{a}_{11}{v}_{1}+{a}_{12}{v}_{2}+\dots +{a}_{1m}{v}_{\mathrm{m}}\\ {a}_{21}{v}_{1}+{a}_{22}{v}_{2}+\dots +{a}_{2m}{v}_{\mathrm{m}}\\ \vdots \\ {a}_{n1}{v}_{1}+{a}_{n2}{v}_{2}+\dots +{a}_{nm}{v}_{\mathrm{m}}\end{array}\right)$

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