7.3 Properties of Matrices The objects of study
in linear algebra are linear operators. We have
seen that linear operators can be represented
as matrices through choices of ordered bases,
and that mat
right hand side of the above equation really
stand for the vector obtained by multiplying
the coefficients stored in the column vector by
the corresponding basis element and then
summing over them. Ne
scalars r and s). Your answer should have two
parts. Show that (1) (2), and then show that
(2) (1). 118 6.5 Review Problems 119 2. If f is
a linear function of one variable, then how
many points on th
powerful tool for calculations involving linear
transformations. It is important to understand
how to find the matrix of a linear
transformation and the properties of matrices.
7.1 Linear Transformati
w) 6= (u v) w. 149 150 Matrices (b) We saw
in Chapter 1 that the operator B = u (cross
product with a vector) is a linear operator. It
can therefore be written as a matrix (given an
ordered basis such
as the set R S ? Generalize to other size
matrices. 8. Show that any function in R
cfw_,?,# can be written as a sum of multiples of
the functions e, e?, e# defined by e(k) =
1 , k = 0 , k = ? 0 , k
We are now ready to learn the powerful
consequences of linearity. 111 112 Linear
Transformations 6.1 The Consequence of
Linearity Now that we have a sufficiently
general notion of vector space it is t
on any vector by first expressing the vector as
a 116 6.4 Bases (Take 1) 117 sum of multiples
and then applying linearity; L x y = L x + y 2 1
1 + x y 2 1 1 = x + y 2 L 1 1 + x y 2 L 1 1
= x + y 2 2 4
range is the line through the origin in the x2
direction. It is not clear how to formulate L as
a matrix; since L c1 c1 + c2 c2 = 0 0 0
1 0 1 0 0 0 c1 c1 + c2 c2 = (c1 + c2)
0 1 0 , or L c1 c1 + c2
already try to write down the standard basis
vectors for R n for other values of n and
express an arbitrary vector in R n in terms of
them. The last example probably seems
pedantic because column vect
problem 4 There are many ways to cut up an
n n matrix into blocks. Often context or the
entries of the matrix will suggest a useful way
to divide the matrix into blocks. For example,
if there are larg
section 7.3. (b) Multiply out MN and write out
a few of its entries in the same form as in part
(a). In terms of the entries of M and the
entries of N, what is the entry in row i and
column j of MN? (
with standard addition and scalar
multiplication; V := cfw_a0 1 + a1x + a2x 2 | a0,
a1, a2 R Let d dx : V V be the derivative
operator. The following three equations, along
with linearity of the deriv
suppose that M1 exists. Then Mx = 0 x =
M10 = 0. Thus, if M is invertible, then Mx = 0
has no non-zero solutions. On the other hand,
Mx = 0 always has the solution x = 0. If no
other solutions exist,
function q : R R whose graph contains (1, 1),
(3, 2) and (5, 7). (f) second order homogeneous
polynomial function r : R R whose graph
contains (3, 2). (g) number of points required
to specify a third
zero vector in two-dimensional Lorentzian
space-time with zero length. (b) Find and
sketch the collection of all vectors in twodimensional Lorentzian space-time with zero
length. (c) Find and sketch t
two products (doors, and door frames).
Explain how it can be viewed as a function
mapping one vector space into another. (d)
Assuming that L is linear and Lf is 1 door and 2
frames, and Lg is 3 doors
the target space basis. More carefully, if L is a
linear operator from V to W then the matrix
for L in the ordered bases B = (b1, b2, . . .) for V
and B0 = (1, 2, . . .) for W, is the array of
numbers
Associativity of matrix multiplication. We
know for real numbers x, y and z that x(yz) =
(xy)z , i.e., the order of multiplications does
not matter. The same property holds for
matrix multiplication,
0 0 1 Application: Cryptography A very
simple way to hide information is to use a
substitution cipher, in which the alphabet is
permuted and each letter in a message is
systematically exchanged for an
t 0 1 . Then M2 = 1 2t 0 1 , M3 = 1 3t 0 1 , . . .
and so f(M) = 1 t 0 1 2 1 2t 0 1 + 3 1 3t 0 1 = 2
6t 0 2 . Suppose f(x) is any function defined by
a convergent Taylor Series: f(x) = f(0) + f 0 (0)x
M = LU , where: L is lower triangular . This
means that all entries above the main diagonal
are zero. In notation, L = (l i j ) with l i j = 0 for
all j > i. L = l 1 1 0 0 l 2 1 l 2 2 0 l
3 1 l 3 2 l
Suppose L : V linear V and L(v1) = v1 + v2
, L(v2) = 2v1 + v2 . 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
(e) Compare and contrast your results from
parts (b) and (d). 4. Find the matrix for d dx
acting on the vector space of all power series
in the ordered basis (1, x, x2 , x3 , .). Use this
matrix to fi
since (f + g)(2) = 0. 5.2 Other Fields Above, we
defined vector spaces over the real numbers.
One can actually define vector spaces over any
field. This is referred to as choosing a different
base fie
row, a web browser downloading the file can
start displaying an incomplete version of the
picture before the download is complete.
Finally, a compression algorithm is applied to
the matrix to reduce t
AAT 1 (b) Show that the matrix B above is a
right inverse for A, i.e., verify that AB = I . (c) Is
BA defined? (Why or why not?) (d) Let A be an
n m matrix with n > m. Suggest a formula for
a left inv
relatively simple when working over the
complex numbers. This phenomenon occurs
when diagonalizing matrices, see chapter 13.
The rational numbers Q are also a field. This
field is important in compute
directions available. To figure out the
dimension of a vector space, I stand at the
origin, and pick a direction. If there are any
vectors in my vector space that arent in that
direction, then I choos