Lecture 25, March 15, 2017.
De▯nition. A transformation T : R ! R m is called linear if
(1) T(u + v) = T(u) + T(v) for all vectors u;v in R ;
(2) T(cv) = cT(v) for all scalars c and vectors v in R .
Example. Consider the above transformation T : R ! R given by
[ ] ▯ ▯
1 ▯! Av = ▯ 2v1▯ v2 ▯:
3v1+ 4v 2
Let us check that it is linear. Indeed, let u;v be two vectors in R .
Then T(u + v) = A(u + v) = Au + Av = T(u) + T(v). Similarly,
T(cv) = A(cv) = c(Av) = cT(v).
In the above example there is nothing special in size 3 ▯ 2 of the
matrix A. The same is true in general.
Theorem. Let A be a matrix of size m ▯ n. Then the transformation
TA: R ! R m given by v ! TA(v) = Av is a linear transformation.
Example. Let F : R ! R be the transformation that sends each
point to its re
ection in the x-axis. Let us show that it is a linear
transformation. By construction, the transformation F sends the point
(x;y) to the point (x;▯y). Thus we may write
([ ]) [ ]
F y = ▯y :
We could check that F is linear by verifying all conditions in the de▯-
nition. B[t it ]s fa[ter]to o[serve]tha[ ][ ]
x 1 0 1 0 x
= x + y = :
▯y 0 ▯1 0 ▯1 y
Thus ([ ]) [ ]
F y = A y
where [ ]
A = 0 ▯1 :
It now follows from the above theorem that F is linear.
Let us show that any linear transformation T : R ! Ris a matrix
multiplication. Given such T we attach the matrix
A = [T(e )T(e ):::T(e )]
1 2 n
of size m ▯ n where 1 2e ;:::ne is the standard basis.
Theorem. Let T be a linear transformation as above. Then T(v) = Av
for each vector v in R .
Proof. Let ▯ ▯
▯ v2 ▯
v = ▯ . ▯ :
Clearly we have v = v1 1+ v2 2+ ▯▯▯ + vn n. Then with the use of
linearity of T we have
T(v) = T(v 1 1 v 2 2 ▯▯▯ + vn n) = v1T(e1) + ▯▯▯ +nv Tne ) = Av
and we are done. ▯ De▯nition. The above matrix A is called the standard matrix of the
linear transformation T.
Example. Let T : R ! R be the transformation that projects a
point onto x-axis. Find the standard matrix of T.
Solution. This transformation sends a point (x;y) to (x;0). Hence
([ 1 ]) [ 1 ]
T(e1) = T =
([ ]) [ ]
T(e 2 = T 0 = 0 :
Hence the required matrix is