Chapter 2

Question 1

in the transformation y = Ax, what is A?

Answer 1

the (coefficient) matrix of the transformation

Question 2

identity matrix

Answer 2

denoted by I sub n, it’s the n by n matrix whose main diagonal consists of 1s, and the rest is 0s; it’s the coefficient of the identity transformation

Question 3

what makes a linear transformation linear?

Answer 3

T(v + w) = T(v) + T(w) for all vectors v and w in R^m
T(kv) = kT(v) for all vectors v in R^m and all scalars k
T(0) = 0

Question 4

distribution vector

Answer 4

a vector organized such that its components add up to 1 and are all positive or 0

Question 5

transition/stochastic matrix

Answer 5

a square matrix organized such that all of its columns are distribution vectors - all of its entries are positive or zero and every column sums to 2.

Question 6

scaling

Answer 6

a linear transformation that changes the length of a vector. For any positive constant k, the matrix

[k 0
0 k]

defines a scaling by k. When k > 1, it’s a dilation. When k < 1, it’s a contraction.

Question 7

orthogonal projection

definition

Answer 7

the shadow that the vector casts on some line running through the origin if a light were shined right on that line, denoted by projL(x).

Question 8

orthogonal projection

components

Answer 8

the vector x is the sum of that shadow and a line perpendicular to that shadow completing a triangle to it

x = projL(x) + perpendicular(x)

Question 9

orthogonal projection
equation and
transformation matrix

Answer 9

if w is a nonzero vector parallel to L, then
projL(x) = [(x * w)/(w * w)]w

if u is a unit vector in R^2 parallel to L, then
projL(x) = (x * u) * u.

The transformation T(x) = projL(x) is linear with matrix 
P = 1/(w1^2 = w2^2) *
[w1^2  w1w2
w1w2  w2^2] =
[u1^2 u1u2
u1u2 u2^2]

Question 10

reflection

Answer 10

denoted by refL(x), is the reflection about some line L running through the origin.

x = (some line parallel to L) + (perpendicular line connection that parallel line and x)
so
refL = (that parallel line) - (that perpendicular line connecting that parallel line and x)

matrix is
[a b
b -a] where a^2 + b^2 = 1. Any matrix of this form represents a reflection about some line.

Question 11

formula relating reflections with orthogonal projections

Answer 11

refL(x) = 2projL(x) - x = 2(x * u)u - x

Question 12

orthogonal projections and reflections in space

Answer 12

if the plane is perpendicular to a line L, then
projV(x) = x - projL(x) = x - (x * u) * u
refL(x) = projL(x) - projV(x) = 2projL(x) - x = 2(x * u) * u - x
refV(x) = projV(x) - projL(x) = -refL(x) = x - 2(x * u) * u

Question 13

rotation

Answer 13

transformation that rotates a vector x through a fixed angle Z (expressed in polar terms) in the counterclockwise direction.

The matrix in R^2 through an angle Z is
[ cosZ -sinZ
sinZ cosZ ]

it’s of the form
[a -b
b a] where a^2 + b^2 = 1.

Any matrix of this form represents a rotation.

Question 14

rotations combined with a scaling

Answer 14

if r and Z are the polar coordinates of vector [a b] then
[a -b
b a]
represents a rotation through Z combined with a scaling by r.

[a
b] =
[r cos Z
r sinZ ].

r = sqrt(a^2 + b^2)

so Z = arccos[a / sqrt(a^2 + b^2)] and so forth

Question 15

shear

Answer 15

transformation that sort of changes the reference angle of the axis that the shear occurs. Vertical shears change the directioning of things depending on how horizontal they are, and horizontal shears change the directioning of vertical things.

the matrix of a horizontal shear is of the form
[1 k
0 1]
and vertical,
[1 0
k 1], 

where k is an arbitrary constant reflecting the magnitude of the shear somehow, the slope of the “ruler” after transformation.

Question 16

matrix multiplication by column

Answer 16

BA = B[v1 v2 v3 … vm] where vx is a column, of A.

= [Bv1 Bv2 … B*vm]

Question 17

matrix multiplication by the entries of the matrices

Answer 17

the ijth entry of BA is the dot product of the ith row of B with the jth row of column A.

Question 18

five properties of matrix multiplication

Answer 18

• it’s associative
• it’s usually noncommutative
• A(C + D) = AC + AD and (A + B)C = AC + BC
• (kA)B = A(kB) = k(AB)
• A(identity matrix) = identitymatrix(A) = A

Question 19

invertibility and rref

Answer 19

a n x m matrix A is invertible if (and only if) rref(A) = In

or, equivalently, rank(A) = n

A^-1A = I underscore n = AA^-1

Question 20

invertibility and linear systems

Answer 20

If a square matrix A is invertible, then the system Ax = b has the unique solution x = A^-1b. If A is noninvertible then the system Ax = b has infinitely many solutions or none.

when b = 0, the system has x = 0 as a solution. But depending on the invertibility of A it either has one or infinitely many solutions.

Question 21

the inverse of a product of matrices

Answer 21

(BA)^-1 = (A^-1)B^-1

Question 22

what follows from BA = I sub n

Answer 22

A and B are both invertible
A^-1 = B and B^-1 = A
AB = I sub n

Question 23

inverse and determinant of a 2 x 2 matric

Answer 23

A =
[a b
c d]

is invertible iff
ad - bc != 0

if it is invertible,
A^-1 = 1/(ad - bc) *
[d -b
-c a]

Question 24

determinant of a 2 x 2 matrix

Answer 24

A =
[a b
c d]

ad - bc is the determinant of A, written det(A)

also, =
[a
c] * |sin Z| *
[b
d]

it’s the area of the parallelogram spanned by those two columns of A.

If those two columns are parallel, det A = 0
If 0 < Z < pi, det A > 0.
If -pi < Z < 0, det A < 0.