Math 2.0

Linear and multilinear algebra; matrix theory Definition

Scalar multiplication of matrices

Builds on Matrix

Definition

Let RR be a ring, let c∈Rc \in R, and let A=(aij)A = (a_{ij}) be an m×nm \times n matrix over RR. The scalar multiple cAcA is the m×nm \times n matrix with entries (cA)ij=c aij(cA)_{ij} = c\, a_{ij} for 1≤i≤m1 \le i \le m and 1≤j≤n1 \le j \le n.

Remarks

Over a field FF, matrix addition and scalar multiplication make Mm×n(F)M_{m \times n}(F) a vector space of dimension mnmn. The rules c(A+B)=cA+cBc(A + B) = cA + cB, (c+d)A=cA+dA(c + d)A = cA + dA, (cd)A=c(dA)(cd)A = c(dA) and 1A=A1A = A hold. For example, 2(1−103)=(2−206)2 \begin{pmatrix} 1 & -1 \\ 0 & 3 \end{pmatrix} = \begin{pmatrix} 2 & -2 \\ 0 & 6 \end{pmatrix}.

Reading path

To understand this page from the ground up, read these 2 pages in order. The foundations come first.

Show the path as a graph
Matrix, definition, no evidence levelMatrixdefinitionScalar multiplication of matrices, definition, no evidence levelScalar multiplication…definition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Matrix definition
  2. Scalar multiplication of matrices definition

Where this leads

2 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Scalar_multiplication, lead section
About this page
Tag0K28YXMS7Q
URIm2://0K28YXMS7Q
Kinddefinition
WritingW0 stub
MSC15A99
DefinesScalar multiplication of matrices
UsesMatrix

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by