Page 148 - 35Linear Algebra

148 Matrices

   
x 1 y 1
.
.
4. Let x =  .  and y =  .  be column vectors. Show that the
 . 
 . 
x n y n
T
dot product x y = x I y.

Hint

5. Above, we showed that left multiplication by an r × s matrix N was
N
r
s
a linear transformation M −→ M . Show that right multiplication
k
k
R
s
s
by a k × m matrix R is a linear transformation M −→ M . In other
k m
words, show that right matrix multiplication obeys linearity.
Hint

6. Let the V be a vector space where B = (v 1 , v 2 ) is an ordered basis.
Suppose
linear
L : V −−−→ V
and
L(v 1 ) = v 1 + v 2 , L(v 2 ) = 2v 1 + v 2 .

Compute the matrix of L in the basis B and then compute the trace of
this matrix. Suppose that ad − bc 6= 0 and consider now the new basis

0
B = (av 1 + bv 2 , cv 1 + dv 2 ) .

0
Compute the matrix of L in the basis B . Compute the trace of this
matrix. What do you find? What do you conclude about the trace
of a matrix? Does it make sense to talk about the “trace of a linear
transformation” without reference to any bases?

7. Explain what happens to a matrix when:

(a) You multiply it on the left by a diagonal matrix.
(b) You multiply it on the right by a diagonal matrix.

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