Merced College; Don Power

 

LINEAR ALGEBRA - CHAPTER 4, LECTURE

 

4.1  Euclidean n-Space

Hwk:  1c, 4, 5c, 9c, 14cd, 15, 17d, 19, 20, 22, 26 [use u=(a,b), v=(cost,sint)], 30, 34, 37

 

Most properties in this lesson are simple extensions of what we did in R2 and R3 to higher dimensional spaces

      Addition and subtraction

      Dot product (now called Euclidean inner product)

      Norm and distance

      Dot product as a test for orthogonality

      Projections; vector componentsof u along a and orthogonal to a

 

New:

      Cauchy-Schwarz inequality:  abs value of dot product <= product of norms

            Proof in 2D and 3D:  law of cosines

            Proof in higher dimensions:  later

            Application:      prove triangle inequality:  norm of sum <= sum of norms

                                    prove Pythag thm: 

 

4.2  Linear Transformations from Rn to Rm

Hwk:  1bc, 3, 5b, 6bc, 7b, 9b, 10b, 11b, 12b, 13a, 14b, 16b, 18b, 19a, 20ab, 22a, 23 (x-axis), 24, 27, 29b, 34

 

Vocabulary:
function

image (value)

domain, codomain, and range (range may not be the entire codomain:  it is a subset of codomain)

transformations:  function or "map" from Rn to Rm, in symbols f: Rn-->Rm

operator:  domain and codomain are the same, and we write  f: Rn-->Rn

transformation equations:  m equations in n unknowns

      write as vector equation w = Ax, where w is in Rm and x in in Rn

standard matrix Amxn

linear transformation T(x1, x2, ..., xn) = (w1, w2, ..., wm),

      where each wi is a linear function of x1, x2, ..., xn

relationship between matrix mult and calculating function by substitution:

      w = Ax = T(x) = [T]x = TA(x) = [TA]x

 

Specific linear transformations (derive in class, for R2)

      Reflection

      Projection

      Dilation/contraction ("zoom") (multiplying every coordinate by a constant "zoom factor")

      Rotation (counterclockwise through an angle q)

            Let (x1,x2) = (r cosφ, r sinφ),

(w1,w2) = (r cos(φ+q), r sin (φ+q))

= (r [cosφcosq - sinφsinq],r [sinφcosq +cosφsinq])

= (x1cosq - x2sinq, x1sinq + x2cosq)

=

Compositions of linear transformations are calculated in the reverse order (R to L)

      (T2 o T1 )(x) = [T2][T1]x  is calculated by finding T1(x) first, then T of the result

 

Composition of two linear transformations corresponds to multiplication by the product of their standard matrices

 

4.3  Properties of Linear Transformations from Rn to Rm

Hwk:  1, 2, 4, 5a, 7, 8ab, 10ab, 12be, 14a, 15b, 16a, 17b, 18b, 19b, 20ab, 21, 24, 27

 

Equivalent statements (apply to the transformations from lesson 4.2):

      The transformation is "reversible" -- Mathematically, the transformation has an inverse.

      The transformation is one-to-one

      The matrix is invertible

            (Convenient test:  det(A)¹0)

      The range is Rn

Same as saying that for every w in the codomain, there is an x in the domain such that T(x)=w

Thm: (Alt def of linearity)  A transformation T is linear iff

      a.  T(u+v) = T(u) + T(v), and

      b.  T(cu) = cT(u)

      Proof:  Note that the linearity of T is equivalent to saying that a corresponding matrix A exists

            →:  T linear → T(u+v) = A(u+v) = Au + Av = T(u) + T(v); similar proof for part b.

            ←:  With A defined as A = [T] = [ T(e1) | T(e2) | ... | T(en) ], we can show Ax = T(x)

                        (detail in text)

Standard basis vectors of Rn are the column vectors e1 = , e2 = , e3 = , ..., en =

Note:  For R3, these are the same as the standard unit vectors i, j, k

For a linear transformation, the images of the standard basis vectors are the columns of A:

      A = [T] = [ T(e1) | T(e2) | ... | T(en) ], or Aci = T(ei) for each i, i=1..n

 

      Example, p 393: T = projection onto xy-plane: consider T(e1), T(e2), T(e3) to build matrix A

      Example going the other way:  given a 2´3 matrix A, write out the details of  T(ei) =  Aci for each i

      Example (Ex 6)

 

Geometric interpretation of Eigenvalues, Eigenvectors:

      Recall: Ax = λx for Eigenvalues λ and corresponding Eigenvectors x,

      We can view this as a linear operator TA:Rn→Rn for which T(x) = λx

            If λ is positive and greater than 1, we have a dilation by a factor of λ

            If λ = 1, T is the identity transformation.

            If λ is positive and less than 1, we have a contraction

            If λ is negative, we have a reflection about the origin, with a dilation or contraction

 

4.4  Linear Transformations and Polynomials

Hwk:  1b, 2b, 3b, 4a, 6b, 7a, 9a, 11, 15a

 

 

Return to:  Merced College; Don Power               Updated 03/09/07 by Don Power