A matrix elements 1 2 -1 2 3 0 4 -2 2 3 3 5 Elements of a vector of b or c 2 1 -3

 The result of multiplying the vector by the matrix b * A The result of multiplying the matrix by the vector A * b

Each vector can be considered as a single-column or single-line matrix. A single-column matrix will be called a column vector, and a single-row matrix will be called a row vector.

If the A-matrix is ​​of size m * n, then the column vector b has size n, and the row vector b has size m.

Thus, in order to multiply the matrix by a vector, we must consider the vector as a column vector. When multiplying a vector by a matrix, it must be considered as a row vector.

Example.

Multiply matrix

$$\begin{pmatrix} 1+2i & 2+i & 1+3i \\ 2 & 4+2i & 2+5i \end{pmatrix}$$

on complex vector

$$\begin{pmatrix} 2+2i \\ 1+4i \\ 2+2i \end{pmatrix}$$

 The result of multiplying the matrix by the vector A * b $\begin{pmatrix}1+2i & 2+i & 1+3i \\ 2 & 4+2i & 2+5i \\ \end{pmatrix}*\begin{pmatrix}2+2i \\ 1+4i \\ 2+2i \\ \end{pmatrix}=\begin{pmatrix}-578 \\ -1138+44i \\ \end{pmatrix}$ The result of multiplying the vector by the matrix b * A $\begin{pmatrix}2+2i & 1+4i & 2+2i \\ \end{pmatrix}*\begin{pmatrix}1+2i & 2+i \\ 1+3i & 2 \\ 4+2i & 2+5i \\ \end{pmatrix}=\begin{pmatrix}-1370 & -594+28i \\ \end{pmatrix}$

As you can see, with the constant dimension of the vector, we can have two solutions.
I would like to draw your attention to the fact that the matrix in the first and second variant, despite the same values, is completely different (have different dimensions)

In the first case, the vector is considered as a column, and then it is necessary to   multiply the matrix by the vector , and in the second case we have a row vector and then we have the product of the vector by the matrix.

This bot multiplies including vectors and matrices that have complex values. Based on a more comprehensive multiplication matrix calculator  with complex values ​​online

Properties of matrix-vector multiplication

$A$ - the matrix
$b$ - vector column
$c$ - string vector
$\lambda$ - arbitrary number
1. The product of the matrix by the sum of the column vectors is equal to the sum of the products of the matrix by each of the vectors
$A(b_1+b_2)=Ab_1+Ab_2$
2. The product of the sum of row vectors by a matrix is ​​equal to the sum of products of vectors by a matrix
$(c_1+c_2)A=c_1A+c_2A$
3. The common factor of the vector can be moved outside the product of the matrix by vector / vector by matrix
$(\lambda{c})A=\lambda{(cA)}\\A(\lambda{b})=\lambda(Ab)$
4. The product of the row vector by the product of the matrix and the column vector is equivalent to the product of the product of the row vector by the matrix and the column vector.
$c(Ab)=(cA)b$

Good calculations !!

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