quiz 3 Flashcards
shear transformation
let A = [1 2]
[0 1]
the transformation T from IR^2 –> IR^2 defined by T(x) = A(x) is called a shear transformation
what does a sheer transformation look like?
it stretches the area of the matrix (the sheep is being pulled)
dilation / contraction
given a scalar r, define T. IR^2 –> IR^2 by T(x) = rx (multiplying the values by a constant doesn’t change anything but the length of the arrow)
dilation criteria
r > 1
contraction criteria
0 <= r <= 1
when is a set of vectors linearly independent?
when it only has the trivial solution
when is a set of vectors linearly dependent?
if there exists weights (c values) that are not all 0 so that the linear combination of the vectors equals 0
when two vectors are multiples of each other, they are _______
linearly dependent
if a set contains the zero vector, then the set is _________
linearly dependent
in the equation Ax = b, what is A and what does A do?
A is a matrix, and it acts on or transforms vector x to produce a new vector b
a transformation T from IR^n to IR^m is ______
a rule that assigns a vector x in IR^n to a vector T(x) in IR^m
domain of T
IR^n (original)
codomain of T
IR^m (new)
T(x)
the image of x
range of T
the set of all images T(x)
a transformation T is linear if:
1) T(u + v) = T(u) + T(v) for all u, v in the domain of T
2) T(cu) = cT(u) for all scalars c and all u in the domain of T
first part of a transformation T being linear
T(u + v) = T(u) + T(v) for all u, v in the domain of T
second part of a transformation T being linear
T(cu) = cT(u) for all scalars c and all u in the domain of T
A(BC) =
(AB)C
A(B+C) =
AB + AC
(B+C)A =
BA + CA
(A^T)^T =
A
(A + B)^T =
A^T + B^T
(rA^T) =
r(A^T) , r scalar
(AB)^T =
B^T * A^T
