Page 63 - Linear Algebra Done Right
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v 1
v k
v n
a 1,k
w 1
. . . The Matrix of a Linear Map 49
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w m a m,k
Note that in the matrix above only the k th column is displayed (and thus With respect to any
the second index of each displayed a is k). The k th column of M(T) choice of bases, the
consists of the scalars needed to write Tv k as a linear combination of matrix of the 0 linear
the w’s. Thus the picture above should remind you that Tv k is retrieved map (the linear map
from the matrix M(T) by multiplying each entry in the k th column by that takes every vector
the corresponding w from the left column, and then adding up the to 0) consists of all 0’s.
resulting vectors.
n
m
If T is a linear map from F to F , then unless stated otherwise you
should assume that the bases in question are the standard ones (where
the k th basis vector is 1 in the k th slot and 0 in all the other slots). If
you think of elements of F m as columns of m numbers, then you can
think of the k th column of M(T) as T applied to the k th basis vector.
3
2
For example, if T ∈L(F , F ) is defined by
T(x, y) = (x + 3y, 2x + 5y, 7x + 9y),
then T(1, 0) = (1, 2, 7) and T(0, 1) = (3, 5, 9), so the matrix of T (with
respect to the standard bases) is the 3-by-2 matrix
1 3
2 5 .
7 9
Suppose we have bases (v 1 ,...,v n ) of V and (w 1 ,...,w m ) of W.
Thus for each linear map from V to W, we can talk about its matrix
(with respect to these bases, of course). Is the matrix of the sum of two
linear maps equal to the sum of the matrices of the two maps?
Right now this question does not make sense because, though we
have defined the sum of two linear maps, we have not defined the sum
of two matrices. Fortunately the obvious definition of the sum of two
matrices has the right properties. Specifically, we define addition of
matrices of the same size by adding corresponding entries in the ma-
trices:
