Page 64 - Linear Algebra Done Right

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a m,1 ... a m,n  Chapter 3. Linear Maps 
b m,n
b m,1
 
a 1,1 + b 1,1 ... a 1,n + b 1,n
 . . 
=  . . . .  .
 
a m,1 + b m,1 ... a m,n + b m,n
You should verify that with this definition of matrix addition,
3.9 M(T + S) =M(T) +M(S)
whenever T, S ∈L(V, W).
Still assuming that we have some bases in mind, is the matrix of a
scalar times a linear map equal to the scalar times the matrix of the
linear map? Again the question does not make sense because we have
not defined scalar multiplication on matrices. Fortunately the obvious
definition again has the right properties. Specifically, we define the
product of a scalar and a matrix by multiplying each entry in the matrix
by the scalar:
   
a 1,1 ... a 1,n ca 1,1 ... ca 1,n
 . .   . . 
c  . . . .  =  . . . .  .
   
a m,1 ... a m,n ca m,1 ... ca m,n
You should verify that with this definition of scalar multiplication on
matrices,
3.10 M(cT) = cM(T)
whenever c ∈ F and T ∈L(V, W).
Because addition and scalar multiplication have now been defined
for matrices, you should not be surprised that a vector space is about
to appear. We need only a bit of notation so that this new vector space
has a name. The set of all m-by-n matrices with entries in F is denoted
by Mat(m, n, F). You should verify that with addition and scalar mul-
tiplication defined as above, Mat(m, n, F) is a vector space. Note that
the additive identity in Mat(m, n, F) is the m-by-n matrix all of whose
entries equal 0.
Suppose (v 1 ,...,v n ) is a basis of V and (w 1 ,...,w m ) is a basis of W.
Suppose also that we have another vector space U and that (u 1 ,...,u p )