Linear Transformations - Stanford University
[Pages:15]Linear Transformations
The two basic vector operations are addition and scaling. From this perspective, the nicest functions are those which "preserve" these operations:
Def: A linear transformation is a function T : Rn Rm which satisfies: (1) T (x + y) = T (x) + T (y) for all x, y Rn (2) T (cx) = cT (x) for all x Rn and c R.
Fact: If T : Rn Rm is a linear transformation, then T (0) = 0.
We've already met examples of linear transformations. Namely: if A is any m ? n matrix, then the function T : Rn Rm which is matrix-vector multiplication
T (x) = Ax
is a linear transformation.
(Wait: I thought matrices were functions? Technically, no. Matrices are literally just arrays of numbers. However, matrices define functions by matrixvector multiplication, and such functions are always linear transformations.)
Question: Are these all the linear transformations there are? That is, does every linear transformation come from matrix-vector multiplication? Yes:
Prop 13.2: Let T : Rn Rm be a linear transformation. Then the function T is just matrix-vector multiplication: T (x) = Ax for some matrix A.
In fact, the m ? n matrix A is
A = T (e1) ? ? ? T (en).
Terminology: For linear transformations T : Rn Rm, we use the word "kernel" to mean "nullspace." We also say "image of T " to mean "range of T ." So, for a linear transformation T : Rn Rm:
ker(T ) = {x Rn | T (x) = 0} = T -1({0}) im(T ) = {T (x) | x Rn} = T (Rn).
Ways to Visualize functions f : R R (e.g.: f (x) = x2) (1) Set-Theoretic Picture.
(2) Graph of f . (Thinking: y = f (x).) The graph of f : R R is the subset of R2 given by: Graph(f ) = {(x, y) R2 | y = f (x)}.
(3) Level sets of f . (Thinking: f (x) = c.) The level sets of f : R R are the subsets of R of the form {x R | f (x) = c},
for constants c R.
Ways to Visualize functions f : R2 R (e.g.: f (x, y) = x2 + y2) (1) Set-Theoretic Picture.
(2) Graph of f . (Thinking: z = f (x, y).) The graph of f : R2 R is the subset of R3 given by: Graph(f ) = {(x, y, z) R3 | z = f (x, y)}.
(3) Level sets of f . (Thinking: f (x, y) = c.) The level sets of f : R2 R are the subsets of R2 of the form {(x, y) R2 | f (x, y) = c},
for constants c R.
Ways to Visualize functions f : R3 R (e.g.: f (x, y, z) = x2 + y2 + z2) (1) Set-Theoretic Picture. (2) Graph of f . (Thinking: w = f (x, y, z).) (3) Level sets of f . (Thinking: f (x, y, z) = c.)
The level sets of f : R3 R are the subsets of R3 of the form {(x, y, z) R3 | f (x, y, z) = c},
for constants c R.
Curves in R2: Three descriptions
(1) Graph of a function f : R R. (That is: y = f (x)) Such curves must pass the vertical line test.
Example: When we talk about the "curve" y = x2, we actually mean to say: the graph of the function f (x) = x2. That is, we mean the set
{(x, y) R2 | y = x2} = {(x, y) R2 | y = f (x)}.
(2) Level sets of a function F : R2 R. (That is: F (x, y) = c)
Example: When we talk about the "curve" x2 + y2 = 1, we actually mean to say: the level set of the function F (x, y) = x2 + y2 at height 1. That is, we mean the set
{(x, y) R2 | x2 + y2 = 1} = {(x, y) R2 | F (x, y) = 1}.
x = f (t) (3) Parametrically:
y = g(t).
Surfaces in R3: Three descriptions
(1) Graph of a function f : R2 R. (That is: z = f (x, y).) Such surfaces must pass the vertical line test.
Example: When we talk about the "surface" z = x2 + y2, we actually mean to say: the graph of the function f (x, y) = x2 + y2. That is, we mean the set
{(x, y, z) R3 | z = x2 + y2} = {(x, y, z) R3 | z = f (x, y)}.
(2) Level sets of a function F : R3 R. (That is: F (x, y, z) = c.)
Example: When we talk about the "surface" x2 + y2 + z2 = 1, we actually mean to say: the level set of the function F (x, y, z) = x2 + y2 + z2 at height 1. That is, we mean the set
{(x, y, z) R3 | x2 + y2 + z2 = 1} = {(x, y, z) R3 | F (x, y, z) = 1}.
(3) Parametrically. (We'll discuss this another time, perhaps.)
Two Examples of Linear Transformations
(1) Diagonal Matrices: A diagonal matrix is a matrix of the form
d1 0 ? ? ? 0
0
D
=
...
d2 ...
??? ...
0
.
0
0 0 ? ? ? dn
The linear transformation defined by D has the following effect: Vectors are... Stretched/contracted (possibly reflected) in the x1-direction by d1 Stretched/contracted (possibly reflected) in the x2-direction by d2 ... Stretched/contracted (possibly reflected) in the xn-direction by dn.
Stretching in the xi-direction happens if |di| > 1. Contracting in the xi-direction happens if |di| < 1. Reflecting happens if di is negative.
(2) Rotations in R2
We write Rot : R2 R2 for the linear transformation which rotates vectors in R2 counter-clockwise through the angle . Its matrix is:
cos sin
- sin cos
.
The Multivariable Derivative: An Example
Example: Let F : R2 R3 be the function
F (x, y) = (x + 2y, sin(x), ey) = (F1(x, y), F2(x, y), F3(x, y)).
Its derivative is a linear transformation DF (x, y) : R2 R3. The matrix of the linear transformation DF (x, y) is:
F1
x
DF
(x,
y)
=
F2 x
F3
x
F1
y
1
F2 y
=
cos(x)
F3
0
y
2
0 . ey
Notice that (for example) DF (1, 1) is a linear transformation, as is DF (2, 3), etc. That is, each DF (x, y) is a linear transformation R2 R3.
Linear Approximation
Single Variable Setting
Review: In single-variable calc, we look at functions f : R R. We write y = f (x), and at a point (a, f (a)) write:
y dy.
Here, y = f (x) - f (a), while dy = f (a)x = f (a)(x - a). So:
f (x) - f (a) f (a)(x - a).
Therefore:
f (x) f (a) + f (a)(x - a).
The right-hand side f (a) + f (a)(x - a) can be interpreted as follows: It is the best linear approximation to f (x) at x = a. It is the 1st Taylor polynomial to f (x) at x = a. The line y = f (a) + f (a)(x - a) is the tangent line at (a, f (a)).
Multivariable Setting
Now consider functions f : Rn Rm. At a point (a, f (a)), we have exactly the same thing:
f (x) - f (a) Df (a)(x - a).
That is:
f (x) f (a) + Df (a)(x - a).
()
Note: The quantity Df (a) is a matrix, while (x - a) is a vector. That is, Df (a)(x - a) is matrix-vector multiplication.
Example: Let f : R2 R. Let's write x = (x1, x2) and a = (a1, a2). Then () reads:
f (x1, x2) f (a1, a2) +
f x1
(a1
,
a2
)
f x2
(a1,
a2)
x1 - a1 x2 - a2
f
f
= f (a1, a2) + x1 (a1, a2)(x1 - a1) + x2 (a1, a2)(x2 - a2).
Tangent Lines/Planes to Graphs
Fact: Suppose a curve in R2 is given as a graph y = f (x). The equation of the tangent line at (a, f (a)) is:
y = f (a) + f (a)(x - a).
Okay, you knew this from single-variable calculus. How does the multivariable case work? Well:
Fact: Suppose a surface in R3 is given as a graph z = f (x, y). The equation of the tangent plane at (a, b, f (a, b)) is:
f
f
z = f (a, b) + (a, b)(x - a) + (a, b)(y - b).
x
y
Note the similarity between this and the linear approximation to f at (a, b).
Tangent Lines/Planes to Level Sets
Def: For a function F : Rn R, its gradient is the vector in Rn given by:
F F
F
F =
, ,...,
.
x1 x2
xn
Theorem: Consider a level set F (x1, . . . , xn) = c of a function F : Rn R. If (a1, . . . , an) is a point on the level set, then F (a1, . . . , an) is normal to the level set.
Corollary 1: Suppose a curve in R2 is given as a level curve F (x, y) = c. The equation of the tangent line at a point (x0, y0) on the level curve is:
F
F
x (x0, y0)(x - x0) + y (x0, y0)(y - y0) = 0.
Corollary 2: Suppose a surface in R3 is given as a level surface F (x, y, z) = c. The equation of the tangent plane at a point (x0, y0, z0) on the level surface is:
F
F
F
x (x0, y0, z0)(x - x0) + y (x0, y0, z0)(y - y0) + z (x0, y0, z0)(z - z0) = 0.
Q: Do you see why Cor 1 and Cor 2 follow from the Theorem?
Composition and Matrix Multiplication
Recall: Let f : X Y and g : Y Z be functions. Their composition is the function g f : X Z defined by
(g f ) = g(f (x)).
Observations: (1) For this to make sense, we must have: co-domain(f ) = domain(g). (2) Composition is not generally commutative: that is, f g and g f are
usually different. (3) Composition is always associative: (h g) f = h (g f ).
Fact: If T : Rk Rn and S : Rn Rm are both linear transformations, then S T is also a linear transformation.
Question: How can we describe the matrix of the linear transformation S T in terms of the matrices of S and T ?
Fact: Let T : Rn Rn and S : Rn Rm be linear transformations with matrices B and A, respectively. Then the matrix of S T is the product AB.
We can multiply an m ? n matrix A by an n ? k matrix B. The result, AB, will be an m ? k matrix:
(m ? n)(n ? k) (m ? k).
Notice that n appears twice here to "cancel out." That is, we need the number of rows of A to equal the number of columns of B ? otherwise, the product AB makes no sense.
Example 1: Let A be a (3 ? 2)-matrix, and let B be a (2 ? 4)-matrix. The product AB is then a (3 ? 4)-matrix.
Example 2: Let A be a (2 ? 3)-matrix, and let B be a (4 ? 2)-matrix. Then AB is not defined. (But the product BA is defined: it is a (4 ? 3)-matrix.)
Two Model Examples
Example 1A (Elliptic Paraboloid): Consider f : R2 R given by
f (x, y) = x2 + y2.
The level sets of f are curves in R2. The level sets are {(x, y) | x2 + y2 = c}. The graph of f is a surface in R3. The graph is {(x, y, z) | z = x2 + y2}.
Notice that (0, 0, 0) is a local minimum of f .
Note
that
f x
(0,
0)
=
f y
(0,
0)
=
0.
Also,
2f x2
(0,
0)
>
0
and
2f y2
(0,
0)
>
0.
Example 1B (Elliptic Paraboloid): Consider f : R2 R given by f (x, y) = -x2 - y2.
The level sets of f are curves in R2. The level sets are {(x, y) | -x2 - y2 = c}. The graph of f is a surface in R3. The graph is {(x, y, z) | z = -x2 - y2}.
Notice that (0, 0, 0) is a local maximum of f .
Note
that
f x
(0,
0)
=
f y
(0,
0)
=
0.
Also,
2f x2
(0,
0)
<
0
and
2f y2
(0,
0)
<
0.
Example 2 (Hyperbolic Paraboloid): Consider f : R2 R given by
f (x, y) = x2 - y2.
The level sets of f are curves in R2. The level sets are {(x, y) | x2 - y2 = c}. The graph of f is a surface in R3. The graph is {(x, y, z) | z = x2 - y2}.
Notice that (0, 0, 0) is a saddle point of the graph of f .
Note
that
f x
(0,
0)
=
f y
(0,
0)
=
0.
Also,
2f x2
(0,
0)
>
0
while
2f y2
(0,
0)
<
0.
General Remark: In each case, the level sets of f are obtained by slicing the graph of f by planes z = c. Try to visualize this in each case.
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