Math 2.0

Linear and multilinear algebra; matrix theory Definition

Matrix–vector product

Builds on Matrix

Definition

Let RR be a ring, let A=(aij)A = (a_{ij}) be an m×nm \times n matrix over RR, and let x=(x1,…,xn)T∈Rnx = (x_1, \dots, x_n)^T \in R^n be a column vector. The matrix–vector product AxAx is the column vector in RmR^m with entries (Ax)i=∑j=1naijxj(Ax)_i = \sum_{j=1}^{n} a_{ij} x_j for 1≤i≤m1 \le i \le m.

Remarks

If A1,…,AnA_1, \dots, A_n are the columns of AA, then Ax=x1A1+⋯+xnAnAx = x_1 A_1 + \cdots + x_n A_n, a linear combination of the columns. A system of linear equations is the equation Ax=bAx = b. The map x↦Axx \mapsto Ax is linear, see multiplication by a matrix as a linear map. For example, (1234)(11)=(37)\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \begin{pmatrix} 1 \\ 1 \end{pmatrix} = \begin{pmatrix} 3 \\ 7 \end{pmatrix}.

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Further reading

  1. Web https://en.wikipedia.org/wiki/Matrix_multiplication, lead section
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