Fundamentals of Matrix Algebra, 2011a

Chapter 5
Graphical Exploraons of Vectors
wasn’t linear? 14
Yes, there is. Consider the second part of the linear transformaon definion. It
states that T(k⃗x) = kT(⃗x) for all scalars k. If we let k = 0, we have T(0⃗x) = 0 · T(⃗x), or
more simply, T(⃗0) = ⃗0. That is, if T is to be a linear transformaon, it must send the
zero vector to the zero vector.
This is a quick way to see that the translaon transformaon fails to be linear. By
shiing the unit square to the right one unit, the corner at the point (0, 0) was sent to
the point (1, 0), i.e.,
[ ]
[ ]
0 1
the vector was sent to the vector .
0 0
This property relang to ⃗0 is important, so we highlight it here.
. Key Idea 15 Linear Transformaons and ⃗0
Let T : R n → R m be a linear transformaon. Then:
.
T(⃗0 n ) = ⃗0 m .
That is, the zero vector in R n gets sent to the zero vector in
R m .
The interested reader may wish to read the footnote below. 15
The Standard Matrix of a Linear Transformaon
It is oen the case that while one can describe a linear transformaon, one doesn’t
know what matrix performs that transformaon (i.e., one doesn’t know the standard
matrix of that linear transformaon). How do we systemacally find it? We’ll need a
new definion.
.
Ḋefinion 31
Standard Unit Vectors
.
In R n , the standard unit vectors ⃗e i are the vectors with a 1
in the i th entry and 0s everywhere else.
14 That is, apart from applying the definion directly?
15 The idea that linear transformaons “send zero to zero” has an interesng relaon to terminology. The
reader is likely familiar with funcons like f(x) = 2x + 3 and would likely refer to this as a “linear funcon.”
However, f(0) ≠ 0, so f is not “linear” by our new definion of linear. We erroneously call f “linear” since
its graph produces a line, though we should be careful to instead state that “the graph of f is a line.”
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