Elementary Linear Algebra

1
Elementary Linear Algebra

• Linear Transformations

2
Contents

• General Linear Transformations
• Kernel and Range
• Inverse Linear Transformations
• Matrices of General Linear Transformations
• Similarity
• Isomorphism

3
Linear Transformation

• Definition
• If T V ? W is a function from a vector space V
  into a vector space W, then T is called a linear
  transformation from V to W if for all vectors u
  and v in V and all scalars c
• T (u v) T (u) T (v)
• T (cu) cT (u)
• In the special case where V W, the linear
  transformation T V ? V is called a linear
  operator on V.

4
Linear Transformation

• Example (Zero Transformation)
• The mapping T V ? W such that T(v) 0 for
  every v in V is a linear transformation called
  the zero transformation.
• Example (Identity Operator)
• The mapping I V ? I defined by I (v) v is
  called the identity operator on V.

5
Orthogonal Projections

• Suppose that W is a finite-dimensional subspace
  of an inner product space V then the orthogonal
  projection of V onto W is the transformation
  defined by
• T (v) projWv
• If S w1, w2, , wr is any orthogonal basis
  for W, then T (v) is given by the formula
• T (v) projWv ?v, w1? w1 ?v, w2? w2
  ?v, wr? wr

6
Orthogonal Projections

• This projection a linear transformation
• T(u v) T(u) T(v)
• T(cu) cT(u)

7
A Linear Transformation from a Space V to Rn

• Let S w1, w2, , wn be a basis for an
  n-dimensional vector space V, and let
• (v)s (k1,, k2,, , kn)
• be the coordinate vector relative to S of a
  vector v in V thus v k1w1 k2w2 kn wn

8
A Linear Transformation from a Space V to Rn

• Define T V ? Rn to be the function that maps v
  into its coordinate vector relative to S that
  is,
• T (v) (v)s (k1,, k2,, , kn)
• Then the function T is a linear transformation
• Let u c1w1 c2w2 cn wn and v d1w1
  d2w2 dn wn
• Check if (u v)s (u)s (v)s and (ku)s
  k(u)s

9
A Linear Transformation from Pn to Pn1

• Let p p(x) c0 c1x cnx n be a
  polynomial in Pn , and define the function T Pn
  ? Pn1 by
• T (p) T (p(x)) xp(x) c0x c1x2 cnx
  n1
• The function T is a linear transformation
• For any scalar k and any polynomials p1 and p2 in
  Pn we have
• T (p1 p2) T (p1(x) p2 (x)) x (p1(x) p2
  (x)) x p1(x) x p2 (x) T (p1) T (p2)
• T (k p) T (k p(x)) x (k p(x)) k (x p(x)) k
  T(p)

10
A Linear Transformation Using an Inner Product

• Let V be an inner product space and let v0 be
  any fixed vector in V. Let T V ? R be the
  transformation that maps a vector v into its
  inner product with v0 that is,
• T (v) ?v, v0?
• From the properties of an inner product
• T (u v) ?u v, v0? ?u, v0? ?v, v0?
• T (k u) ?k u, v0? k ?u, v0? kT (u)
• Thus, T is a linear transformation.

11
Example

• Let TMnn ?R be the transformation that maps an n
  n matrix into its determinant that is,
• T (A) det (A)
• If ngt1, then this transformation does not satisfy
  either of the properties required of a linear
  transformation.

12
Example

• For example, we saw Example 1 of Section 2.3 that
• det (A1A2) ? det (A1) det (A2)
• in general.
• Moreover, det (cA) C n det (A), so
• det (cA) ? c det (A)
• in general.
• Thus, T is not linear transformation.

13
Properties of Linear Transformation

• If T V ? W is a linear transformation, then
  for any vectors v1 and v2 in V and any scalars c1
  and c2, we have
• T (c1v1 c2v2) T (c1v1) T (c2v2) c1T (v1)
  c2T (v2)
• More generally, if v1 , v2 , , vn are vectors in
  V and c1 , c2 , , cn are scalars, then
• T (c1v1 c2v2 cnvn ) c1T (v1) c2T (v2)
  cnT (vn)

14
Properties of Linear Transformation

• The above equation is sometimes described by
  saying that linear transformations preserve
  linear combinations.

15
Theorem

• Theorem 8.1
• If T V ? W is a linear transformation, then
• T(0) 0
• T(-v) -T(v) for all v in V
• T(v w) T(v) T(w) for all v and w in V

16
Finding Linear Transformations from Images of
Basis

• If T V ? W is a linear transformation, and if
  v1 , v2 , , vn is any basis for V, then the
  image T (v) of any vector v in V can be
  calculated from the images
• T (v1), T (v2), , T (vn)
• of the basis vectors.
• This can be done by first expressing v as a
  linear combination of the basis vectors, say
• v c1 v1 c2 v2 cn vn
• and then the transformation becomes
• T (v) c1 T (v1) c2 T (v2) cn T (vn)
• A linear transformation is completely determined
  by its images of any basis vectors.

17
Example

• Consider the basis S v1 , v2 , v3 for R3 ,
  where
• v1 (1,1,1), v2 (1,1,0), and v3 (1,0,0).
• Let T R3 ? R2 be the linear transformation
  such that
• T (v1) (1,0), T (v2) (2,-1), T (v3) (4,3).
• Find a formula for T (x1, x2, x3) then use this
  formula to compute T (2, -3, 5).

18
Example

• Solution
• Let x c1v1 c2v2 c3v3, or (x1, x2 , x3) c1
  (1,1,1) c2 (1,1,0) c3 (1,0,0), then we have
• c1 c2 c3 x1
• c1 c2 x2
• c1 x3
• which yields c1 x3 , c2 x2 x3 , c3
  x1 x2
• Thus, x (x1, x2, x3) x3(1,1,1) (x2
  x3)(1,1,0) (x1 x2)(1,0,0)
• x3v1 (x2 x3 )v2 (x1 x2 )v3
• That is, T (x) T (x1, x2, x3) x3 T (v1) (x2
  x3) T (v2) (x1 x2) T (v3)
• x3(1,0) (x2 x3)(2,-1)
  (x1 x2)(4,3) (4x1 2x2 x3 , 3x1 4x2
  x3)
• From this formula we obtain T (2 , -3 , 5 ) (9,
  23).

19
Composition of T2 with T1

• Definition
• If T1 U ? V and T2 V ? W are linear
  transformations, the composition of T2 with T1,
  denoted by T2 ? T1 (read T2 circle T1 ), is
  the function defined by the formula
• (T2 ? T1 )(u) T2 (T1 (u))
• where u is a vector in U.

20
Composition of T2 with T1

• Theorem 8.1.2
• If T1 U ? V and T2 V ? W are linear
  transformations, then (T2 ? T1 ) U ? W is
  also a linear transformation.

21
Remark

• The compositions can be defined for more than two
  linear transformations.
• For example, if T1 U ? V and T2 V ? W ,and
  T3 W ? Y are linear transformations, then the
  composition T3 ? T2 ? T1 is defined by (T3 ?
  T2 ? T1 )(u) T3 (T2 (T1 (u)))

22
Kernel and Range

• Recall
• If A is an m?n matrix, then the nullspace of A
  consists of all vector x in Rn such that Ax 0.
• The column space of A consists of all vectors b
  in Rm for which there is at least one vector x in
  Rn such that Ax b.
• The nullspace of A consists of all vectors in Rn
  that multiplication by A maps into 0. (in terms
  of matrix transformation)
• The column space of A consists of all vectors in
  Rm that are images of at least one vector in Rn
  under multiplication by A. (in terms of matrix
  transformation)

23
Kernel and Range

• Definition
• If T V ? W is a linear transformation, then
  the set of vectors in V that T maps into 0 is
  called the kernel of T it is denoted by ker(T).
• The set of all vectors in W that are images
  under T of at least one vector in V is called the
  range of T it is denoted by R(T).

24
Examples

• If TA Rn ? Rm is multiplication by the m?n
  matrix A, then the kernel of TA is the nullspace
  of A and the range of TA is the column space of
  A.
• Let T V ? W be the zero transformation. Since T
  maps every vector in V into 0, it follows that
  ker(T) V. Moreover, since 0 is the only image
  under T of vectors in V, we have R(T) 0.
• Let I V ? V be the identity operator. Since I
  (v) v for all vectors in V, every vector in V
  is the image of some vector thus, R(I) V.
  Since the only vector that I maps into 0 is 0, it
  follows ker(I) 0.

25
Example

• Let T R3 ? R3 be the orthogonal projection on
  the xy-plane. The kernel of T is the set of
  points that T maps into 0 (0,0,0) these are
  the points on the z-axis.
• Since T maps every points in R3 into the
  xy-plane, the range of T must be some subset of
  this plane. But every point (x0 ,y0 ,0) in the
  xy-plane is the image under T of some point. Thus
  R(T) is the entire xy-plane.

26
Example

• Let T R2 ? R2 be the linear operator that
  rotates each vector in the xy-plane through the
  angle ?.
• Since every vector in the xy-plane can be
  obtained by rotating through some vector through
  angle ?, we have R(T) R2.
• The only vector that rotates into 0 is 0, so
  ker(T) 0.

27
Properties of Kernel and Range

• Theorem 8.2.1
• If T V ? W is linear transformation, then
• The kernel of T is a subspace of V.
• The range of T is a subspace of W.
• Definition
• If T V ? W is a linear transformation, then the
  dimension of the range of T is called the rank of
  T and is denoted by rank(T).
• The dimension of the kernel is called the nullity
  of T and is denoted by nullity(T).

28
Properties of Kernel and Range

• Theorem 8.2.2
• If A is an m?n matrix and TA Rn ? Rm is
  multiplication by A, then
• nullity (TA) nullity (A)
• rank (TA) rank (A)

29
Example

• Let TA R6 ? R4 be multiplication byFind
  the rank and nullity of TA
• In Example 1 of Section 5.6 we showed that rank
  (A) 2 and nullity (A) 4. (use reduced
  row-echelon form, etc.)
• Thus, from Theorem 8.2.2, rank (TA) 2 and
  nullity (TA) 4.

30
Example

• Let T R3 ? R3 be the orthogonal projection on
  the xy-plane.
• From Example 4, the kernel of T is the z-axis,
  which is one-dimensional and the range of T is
  the xy-plane, which is two-dimensional.
• Thus, nullity (T) 1 and rank (T) 2.

31
Dimension Theorem for Linear Transformations

• Theorem 8.2.3
• If T V ? W is a linear transformation from an
  n-dimensional vector space V to a vector space W,
  then
• rank(T) nullity(T) n
• Remark
• In words, this theorem states that for linear
  transformations the rank plus the nullity is
  equal to the dimension of the domain.

32
Dimension Theorem for Linear Transformations

• Example
• Let T R2 ? R2 be the linear operator that
  rotates each vector in the xy-plane through an
  angle ?. We showed that ker(T) 0 and R(T)
  R2.
• Thus, rank(T) nullity(T) 2 0 2.

33
One-to-One Linear Transformation

• Definition
• A linear transformation T V ? W is said to be
  one-to-one if T maps distinct vectors in V into
  distinct vectors in W.

34
One-to-One Linear Transformation

• Examples
• If A is an n?n matrix and TA Rn ? Rn is
  multiplication by A, then TA is one-to-one if and
  only if A is an invertible matrix (Theorem
  4.3.1).
• Let T Pn ? Pn1 be the linear transformation T
  (p) T(p(x)) xp(x). If p p(x) c0 c1 x
  cn xn and q q(x) d0 d1 x dn xn are
  distinct polynomials, then they differ in at
  least one coefficient. Thus, T (p) c0 x c1
  x2 cn xn1 and T (q) d0 x d1 x2 dn
  xn1 also differ in at least one coefficient.
  Thus, T is one-to-one, since it maps distinct
  polynomials p and q into distinct polynomials T
  (p) and T (q).

35
Theorems

• Theorem 8.3.1 (Equivalent Statements)
• If T V ? W is a linear transformation, then the
  following are equivalent.
• T is one-to-one
• The kernel of T contains only zero vector that
  is, ker(T) 0
• Nullity(T) 0

36
Theorems

• Theorem 8.3.2
• If V is a finite-dimensional vector space and T
  V ? V is a linear operator, then the following
  are equivalent.
• T is one-to-one
• ker(T) 0
• Nullity(T) 0
• The range of T is V that is, R(T) V

37
Example

• Let TA R4 ? R4 be multiplication
  byDetermine whether TA is one to one.
• Solution
• det(A) 0, since the first two rows of A are
  proportional ? A is not invertible? TA is not
  one-to-one.

38
Inverse Linear Transformations

• If T V ? W is a linear transformation, then the
  range of T denoted by R (T), is the subspace of W
  consisting of all images under T of vectors in V.
  
• If T is one-to-one, then each vector w in R(T) is
  the image of a unique vector v in V.
• This uniqueness allows us to define a new
  function, call the inverse of T, denoted by T 1,
  which maps w back into v.
• The mapping T 1 R (T) ? V is a linear
  transformation. Moreover,
• T 1(T (v)) T 1(w) v
• T 1(T (w)) T 1(v) w

39
Inverse Linear Transformations

• If T V ? W is a one-to-one linear
  transformation, then the domain of T 1 is the
  range of T.
• The range may or may not be all of W (one-to-one
  but not onto).
• For the special case that T V ? V, then the
  linear transformation is one-to-one and onto.

40
Example (An Inverse Transformation)

• Let T R3 ? R3 be the linear operator defined by
  the formula
• T (x1, x2, x3) (3x1 x2, -2x1 4x2 3x3, 5x1
  4 x2 2x3).
• Determine whether T is one-to-one if so, find T
  -1(x1,x2,x3) .

41
Example (An Inverse Transformation)

• Solution

42
Theorem

• Theorem 8.3.3
• If T1 U ? V and T2 V ? W are one to one
  linear transformation then
• T2 ? T1 is one to one
• (T2 ? T1)-1 T1-1 ? T2-1

43
Matrices of General Linear Transformations

• Remark
• If V and W are finite-dimensional vector spaces
  (not necessarily Rn and Rm), then any
  transformation T V ? W can be regarded as a
  matrix transformation.
• The basic idea is to work with coordinate
  matrices of the vectors rather than with the
  vectors themselves.

44
Matrices of Linear Transformations

• Suppose V and W are n and m dimensional vector
  space and B and B? are bases for V and W, then
  for x in V, the coordinate matrix xB will be a
  vector in Rn, and coordinate matrix T(x) B?
  will be a vector in Rm .

45
Matrices of Linear Transformations

• If we let A be the standard matrix for this
  transformation, then A xB T (x)B?
• The matrix A is called the matrix for T with
  respect to the bases B and B?

46
Matrices of Linear Transformations

• Let B u1, , un be a basis for the
  n-dimensional space V and B? u1, , um be a
  basis for the m-dimensional space W.
• Consider an m?n matrix such that A xB
  T(x)B? holds for all vectors x in V.

47
Matrices of Linear Transformations

• That is, A xB T(x)B? has to hold for the
  basis vectors u1, , un.
• Thus, we need
• A u1B T(u1)B? ,
• A u2B T(u2)B? , , A unB T(un)B?
• Since
• u1B e1 , u2B e2 , , unB en

48
Matrices of Linear Transformations

• We have
• Thus, , which is the matrix for T w.r.t. the
  bases B and B?, and denoted by the symbol TB?,B

49
Matrices of Linear Transformations

• That is,
• and

Basis for the image space
Basis for the domain
50
Matrices for Linear Operators

• In the special case where V W, the resulting
  matrix is called the matrix for T with respect to
  the basis B and denoted by TB rather than
  TB,B.
• If B u1, , un , then we have
• and
• That is, the matrix for T times the coordinate
  matrix for x is the coordinate matrix for T(x).

51
Example

• Let T P1 ? P2 be the transformations defined
  by
• T (p(x)) xp(x).
• Find the matrix for T with respect to the
  standard bases,
• B u1, u2 and B? v1, v2, v3,
• where u1 1, u2 x v1 1, v2 x , v3 x2
• Solution
• T(u1) T(1) (x)(1) x and T(u2)
  T(x) (x)(x) x2
• T (u1)B 0 1 0T T (u2)B 0 0 1T
• Thus, the matrix for T w.r.t. B and B is

52
Example

• Let T R2 ? R3 be the linear transformation
  defined by
• Find the matrix for the transformation T with
  respect to the bases B u1,u2 for R2 and B?
  v1,v2,v3 for R3, where
• Solution

53
Example
54
Theorems

• Theorem 8.4.1
• If T Rn ? Rm is a linear transformation and if
  B and B? are the standard bases for Rn and Rm,
  respectively, then
• TB?,B T
• Theorem 8.4.2
• If T1 U ? V and T2 V ? W are linear
  transformations, and if B, B?? and B? are bases
  for U, V and W, respectively, then
• T2 ? T1B,B T2 B,BT1 B,B
• Theorem 8.4.3
• If T V ? V is a linear operator and if B is a
  basis for V then the following are equivalent
• T is one to one
• TB is invertible
• Moreover, when these equivalent conditions hold
• T-1B TB-1

55
Indirect Computation of a Linear Transformation

• An indirect procedure to compute a linear
  transformation
• Compute the coordinate matrix xB
• Multiply xB on the left by TB?,B to produce
  T (x)B?
• Reconstruct T (x) from its coordinate matrix T
  (x)B?

56
Example

• Let T P2 ? P2 be linear operator defined by
  T(p(x)) p(3x 5), that is, T (co c1x c2x2)
  co c1(3x 5) c2(3x 5)2
• Find TB with respect to the basis B 1, x,
  x2
• Use the indirect procedure to compute T (1 2x
  3x2)
• Check the result by computing T (1 2x 3x2)

57
Example

• Solution
• Form the formula for T,
• T(1) 1, T(x) 3x 5, T(x2) (3x 5)2 9x2
  30x 25
• Thus,

58
Example

• The coordinate matrix relative to B for vector p
  1 2x 3x2 is
• pB 1 2 3T.
• Thus, T (1 2x 3x2)B T (p)B TB pB
  ? T (1 2x 3x2) 66 84x 27x2
• By direction computation
• T (1 2x 3x2) 1 2(3x 5) 3(3x 5)2
• 1 6x 10 27x2 90x 75
• 66 84x 27x2

59
Similarity

• The matrix of a linear operator T V ? V depends
  on the basis selected for V that makes the matrix
  for T as simple as possible a diagonal or
  triangular matrix.

60
Simple Matrices for Linear Operators

• Consider the linear operator T R2 ? R2 defined
  by
• and the standard basis B e1, e2 for R2.
• The matrix for T with respect to this basis is
  the standard matrix for T that is, TB T
  T(e1) T(e2).
• Since T (e1) 1 -2T, T (e2) 1 4T, we have

61
Simple Matrices for Linear Operators

• However, if u1 1 1T, u2 1 2T, then the
  matrix for T with respect to the basis B? u1,
  u2 is the diagonal matrix
• This matrix is simpler in the sense that
  diagonal matrices enjoy special properties that
  more general matrices do not!

62
Theorem

• Theorem 8.5.1
• If B and B? are bases for a finite-dimensional
  vector space V, and if I V ? V is the identity
  operator, then IB,B? is the transition matrix
  from B? to B.
• Remark

I
V
V
v
v
Basis B?
Basis B
IB,B? is the transition matrix from B? to B.
63
Theorem

• Theorem 8.5.2
• Let T V ? V be a linear operator on a
  finite-dimensional vector space V, and let B and
  B? be bases for V. Then
• TB? P-1 TB P
• where P is the transition matrix from B? to B.
• Remark

I
I
T
v
v
T(v)
T(v)
V
V
V
V
Basis B?
Basis B?
Basis B
Basis B
TB? IB?,BTBIB,B? P-1 TB P
64
Example

• Let T R2 ? R2 be defined byFind the matrix T
  with respect to the standard basis B e1, e2
  for R2, then use Theorem 8.5.2 to find the matrix
  of T with respect to the basis B? u1?, u2?,
  where u1? 1 1T and u2? 1 2T.

65
Example

• Solution
• By inspection, u1? e1 e2 and u2? e1 2 e2,
  ? u1?B 1 1T and u2?B 1 2T.

66
Definitions

• Definition
• If A and B are square matrices, we say that B is
  similar to A if there is an invertible matrix P
  such that B P-1AP
• Definition
• A property of square matrices is said to be a
  similarity invariant or invariant under
  similarity if that property is shared by any two
  similar matrices.

67
Similarity Invariants
68
Determinant of A Linear Operator

• Two matrices representing the same linear
  operator T V ? V with respect to different
  bases are similar.
• For any two bases B and B? we must have
• det(TB) det(TB?)
• Thus we define the determinant of the linear
  operator T to be
• det(T) det(TB)
• where B is any basis for V.

69
Determinant of A Linear Operator

• Example
• Let T R2 ? R2 be defined by

70
Eigenvalues of a Linear Operator

• A scalar l is called an eigenvalue of a linear
  operator T V ? V if there is a nonzero vector x
  in V such that Tx lx. The vector x is called an
  eigenvector of T corresponding to l.
• Equivalently, the eigenvectors of T corresponding
  to l are the nonzero vectors in the kernel of lI
  T. This kernel is called the eigenspace of T
  corresponding to l.

71
Eigenvalues of a Linear Operator

• If V is a finite-dimensional vector space, and B
  is any basis for V, then
• The eigenvalues of T are the same as the
  eigenvalues of TB .
• A vector x is an eigenvector of T corresponding
  to TB if and only if its coordinate matrix xB
  is an eigenvector of TB corresponding to l.

72
Example

• Find the eigenvalues and bases for the
  eigenvalues of the linear operator T P2 ? P2
  defined by
• T (a bx cx2) -2c (a 2b c)x (a
  3c)x2

73
Example

• Solution
• The matrix for T with respect to the standard
  basis B 1, x, x2 is
• The eigenvalues of T are l 1 and l 2
• The eigenvectors of TB are
• l 2 l 1
• Thus, the eigenvectors of T are p1 -1 x2 , p2
  x , p3 -2 x x2
• Check T (p1) 2p1, T (p2) 2p2 and T (p3) p3

74
Example

• Let T R3 ? R3 be the linear operator given
  byFind a basis for R3 relative to which the
  matrix for T is diagonal.
• Solution
• det(lI - A) l3 - 5l2 8l - 4 (l-2)(l-2)(l-1)

75
Onto Transformations

• Definitions
• Let V and W be real vector spaces. We say that
  the linear transformation T V ? W is onto if
  the range of T is W.
• An onto transformation is also said to be
  surjective or to be a surjection. For a
  surjective mapping, the range and the codomain
  coincide.
• If a transformation T V ? W is both one-to-one
  (also called injective or an injection) and onto,
  then it is a one-to-one mapping to its range W
  and so has an inverse T-1 W ? V.

76
Onto Transformations

• Theorem 8.6.1 (Bijective Linear Transformation)
• Let V and W be finite-dimensional vector spaces.
  If dim(V) ? dim(W), then there can be no
  bijective linear transformation from V to W.

77
Isomorphisms

• Definition
• An isomorphism between V and W is a bijective
  linear transformation from V to W.
• Theorem 8.6.2 (Isomorphism Theorem)
• Let V be a finite-dimensional real vector space.
  If dim(V) n, then there is an isomorphism from
  V to Rn.
• Example
• The vector space P3 is isomorphic to R4, because
  the transformation
• T(a bx cx2 dx3) (a,b,c,d)
• is one-to-one, onto, and linear.

78
Isomorphisms between Vector Spaces

• Theorem 8.6.3 (Isomorphism of Finite-Dimensional
  Vector Spaces)
• Let V and W be finite-dimensional vector spaces.
  If dim(V) dim(W), then V and W are isomorphic.

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Isomorphisms between Vector Spaces

• Example (An Isomorphism between P3 and M22)
• Because dim(P3) 4 and dim(M22) 4, these
  spaces are isomorphic.
• We can find an isomorphism T P3 ? M22
• This is one-to-one and onto linear
  transformation, so it is an isomorphism between
  P3 and M22.