linear-guest

148 Matrices
⎛ ⎞ ⎛ ⎞
x 1
y 1
⎜ ⎟ ⎜ ⎟
4. Let x = ⎝ . ⎠ and y = ⎝ . ⎠ be column vectors. Show that the
x n y n
dot product x y = x T I y.
Hint
5. Above, we showed that left multiplication by an r × s matrix N was
a linear transformation Mk
s N
−→ Mk r . Show that right multiplication
by a k × m matrix R is a linear transformation Mk
s R
−→ Mm. s In other
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
L : V −−−→ linear
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 ≠ 0 and consider now the new basis
B ′ = (av 1 + bv 2 , cv 1 + dv 2 ) .
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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