Given two linear transformations [math]S, T:\mathbb{R}^n\to\mathbb{R}^m[/math], we define their sum as follows: For any vector [math]u[/math] in [math]\mathbb{R}^n[/math],[br][br][math](S+T)(u)=S(u)+T(u)[/math][br][br]The definition makes sense because for any vector [math]u[/math] in [math]\mathbb{R}^n[/math], both [math]S(u)[/math] and [math]T(u)[/math] are vectors in [math]\mathbb{R}^m[/math] and hence we can add them together.[br][br]It is easy to see that [math]S+T[/math] is again a linear transformation from [math]\mathbb{R}^n[/math] to [math]\mathbb{R}^m[/math].[br][br]In the previous section, we already learned that any linear transformation [math]T:\mathbb{R}^n\to\mathbb{R}^m[/math] can be uniquely represented by an m x n matrix. What can we say about the relationship among the matrices for [math]S, T[/math] and [math]S+T[/math]?[br][br]First of all, we write down the matrices for [math]S[/math] and [math]T[/math] respectively. Let [math]\{ e_1,e_2,\ldots,e_n\}[/math] be the standard basis for [math]\mathbb{R}^n[/math]. Then [math]S\left(e_1\right),S\left(e_2\right),\ldots,S\left(e_n\right)[/math] are the n columns of the matrix for [math]S[/math]. Similarly, [math]T\left(e_1\right),T\left(e_2\right),\ldots,T\left(e_n\right)[/math] are the n columns of the matrix for [math]T[/math]. The matrices can be expressed as follows:[br][br][math]A=\left[S(e_1) \ S(e_2) \ \cdots \ S(e_n)\right][/math][br][math]B=\left[T(e_1) \ T(e_2) \ \cdots \ T(e_n)\right][/math][br][br]They are m x n matrices because each column is a column vector in [math]\mathbb{R}^m[/math]. Now, we consider the matrix for S+T. It is formed by column vectors [math](S+T)\left(e_1\right),(S+T)\left(e_2\right),\ldots,(S+T)\left(e_n\right)[/math]. By definition, [math](S+T)(e_j)[/math] is simply [math]S(e_j)+T(e_j)[/math] for [math]j=1,\ldots,n[/math]. Therefore, the matrix for [math]S+T[/math] is[br][br][math]\left[S(e_1)+T(e_1) \ S(e_2)+T(e_2) \ \cdots \ S(e_n)+T(e_n)\right][/math][br][br]Therefore, it is natural to define [math]A+B[/math] to be the above m x n matrix. [br][br]More explicitly, we can also write down an m x n matrix entry by entry as follows:[br][br][math]A=\begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ [br]\vdots & \vdots & \ldots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \end{pmatrix}, \ \ \ B=\begin{pmatrix} b_{11} & b_{12} & \cdots & b_{1n} \\ b_{21} & b_{22} & \cdots & b_{2n} \\ [br]\vdots & \vdots & \ldots & \vdots \\ b_{m1} & b_{m2} & \cdots & b_{mn} \end{pmatrix}[/math][br][br]Usually, we can refer to a specific entry in an m x n matrix by the indices in the subscript - [math]a_{ij}[/math] is the entry of the matrix [math]A[/math] on the [math]i^{\text{th}}[/math] row and [math]j^{\text{th}}[/math] column. According to the above definition [math]A+B[/math] can be obtained by adding the corresponding column vectors of [math]A[/math] and [math]B[/math] together, which implies that each entry in [math]A+B[/math] is just the sum of the corresponding entries of [math]A[/math] and [math]B[/math] i.e. we have[br][br][math]A+B=\begin{pmatrix} a_{11}+b_{11} & a_{12}+b_{12} & \cdots & a_{1n}+b_{1n} \\ a_{21}+b_{21} & a_{22}+b_{22} & \cdots & a_{2n}+a_{2n} \\ [br]\vdots & \vdots & \ldots & \vdots \\ a_{m1}+b_{m1} & a_{m2}+b_{m2} & \cdots & a_{mn}+b_{mn} \end{pmatrix}[/math][br][br]This is how we compute the addition of two matrices of the same size. For matrices with different sizes, it does not make sense to define the addition of them.[br]

Given a linear transformation [math]S:\mathbb{R}^n\to\mathbb{R}^m[/math] and a real number [math]k[/math], we define the scaling of [math]S[/math] by [math]k[/math] as follows: For any vector [math]u[/math] in [math]\mathbb{R}^n[/math],[br][br][math](kS)(u)=kS(u)[/math][br][br]Since [math]S(u)[/math] is a vector in [math]\mathbb{R}^m[/math], it makes sense to scale it by [math]k[/math]. Morever, it is easy to see that [math]kS[/math] is also a linear transformation from [math]\mathbb{R}^n[/math] to [math]\mathbb{R}^m[/math]. The matrix for [math]kS[/math] is as follows:[br][br][math]\left[ (kS)(e_1) \ (kS)(e_2) \ \cdots \ (kS)(e_n)\right][/math][br][br]By definition, [math](kS)(e_j)=kS(e_j)[/math] for [math]j=1,\ldots, n[/math]. Therefore, each entry of this matrix is simply k times the corresponding entry of the matrix for [math]S[/math]. Let A be the matrix for S, then it is natural to define kA as follows:[br][br][math]kA=\begin{pmatrix} ka_{11} & ka_{12} & \cdots & ka_{1n} \\ ka_{21} & ka_{22} & \cdots & ka_{2n} \\ [br]\vdots & \vdots & \ldots & \vdots \\ ka_{m1} & ka_{m2} & \cdots & ka_{mn} \end{pmatrix}[/math][br][br]This definition is consistent with the scaling of column vectors if we regard column vectors as n x 1 matrices.[br]

Two matrices are said to be [b]equal[/b] if they have the same size and their corresponding entries are equal.[br][br]From the definitions above, it is not difficult to see that the usual rules algebra apply to sums and scaling of matrices: Let [math]A,B[/math] and [math]C[/math] be matrices of the same size, and let [math]r[/math] and [math]s[/math] be real numbers.[br][br][list=1][*][math]A+B=B+A[/math][br][/*][*][math](A+B)+C=A+(B+C)[/math][br][/*][*][math]A+0=A[/math] (Note: Here "0" means [b]zero matrix[/b] i.e. a matrix with all entries zero)[/*][*][math]r(A+B)=rA+rB[/math][br][/*][*][math](r+s)A=rA+sA[/math][br][/*][*][math]r(sA)=(rs)A[/math][br][/*][/list][br]The following are some questions about matrix algebra: