Sections 6.5 and 7.1
Week 3
Ali Mousavidehshikh
Department of Mathematics University of Toronto
Ali Mousavidehshikh
Week 3
Outline
1 Sections 6.5 and 7.1
Sections 6.5 and 7.1
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
(Theorem)Letp0(x),p1(x),…,pn bepolynomialsinPn of degree 0, 1, . . . , n, respectively. Then
{p0(x), p1(x), . . . , pn(x)} is a basis of Pn.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
(Theorem)Letp0(x),p1(x),…,pn bepolynomialsinPn of degree 0, 1, . . . , n, respectively. Then
{p0(x), p1(x), . . . , pn(x)} is a basis of Pn.
Consequently, for any real number a, {1, x − a, . . . , (x − a)n} is a basis for Pn.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
(Theorem)Letp0(x),p1(x),…,pn bepolynomialsinPn of degree 0, 1, . . . , n, respectively. Then
{p0(x), p1(x), . . . , pn(x)} is a basis of Pn.
Consequently, for any real number a, {1, x − a, . . . , (x − a)n} is a basis for Pn.
(Corollary) If a is any real number, every polynomial f (x) of degree at most n has an expansion in powers of (x − a):
f (x) = a0 + a1(x − a) + . . . + an(x − a)n.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
(Theorem)Letp0(x),p1(x),…,pn bepolynomialsinPn of degree 0, 1, . . . , n, respectively. Then
{p0(x), p1(x), . . . , pn(x)} is a basis of Pn.
Consequently, for any real number a, {1, x − a, . . . , (x − a)n} is a basis for Pn.
(Corollary) If a is any real number, every polynomial f (x) of degree at most n has an expansion in powers of (x − a):
f (x) = a0 + a1(x − a) + . . . + an(x − a)n.
(Corollary) Let f (x) be a polynomial of degree n ≥ 1 and let a be any real number. Then
1. f (x) = f (a) + (x − a)g(x) for some polynomial g(x) of degree n.
Ali Mousavidehshikh
Week 3
Sections 6.5 and 7.1
(Theorem)Letp0(x),p1(x),…,pn bepolynomialsinPn of degree 0, 1, . . . , n, respectively. Then
{p0(x), p1(x), . . . , pn(x)} is a basis of Pn.
Consequently, for any real number a, {1, x − a, . . . , (x − a)n} is a basis for Pn.
(Corollary) If a is any real number, every polynomial f (x) of degree at most n has an expansion in powers of (x − a):
f (x) = a0 + a1(x − a) + . . . + an(x − a)n.
(Corollary) Let f (x) be a polynomial of degree n ≥ 1 and let a be any real number. Then
1. f (x) = f (a) + (x − a)g(x) for some polynomial g(x) of degree n.
2. f(a) = 0 if and only if f(x) = (x −a)g(x) for some polynomial g(x) of degree n − 1.
Ali Mousavidehshikh
Week 3
Sections 6.5 and 7.1
(Taylor’s Theorem) If f (x) is a polynomial of degree n, then
f (x) = n f (k)(a)(x − a)k for any real number a, where
k! k=1
f (k)(a) is the k-th derivative of f evaluated at a and f (0)(a) = f (a).
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
(Taylor’s Theorem) If f (x) is a polynomial of degree n, then
f (x) = n f (k)(a)(x − a)k for any real number a, where
k! k=1
f (k)(a) is the k-th derivative of f evaluated at a and f (0)(a) = f (a).
(Example)
f (x) = 5×3 +10x +2 = 17+25(x −1)+15(x −1)2 +5(x −1)3.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
(Taylor’s Theorem) If f (x) is a polynomial of degree n, then
f (x) = n f (k)(a)(x − a)k for any real number a, where
k! k=1
f (k)(a) is the k-th derivative of f evaluated at a and f (0)(a) = f (a).
(Example)
f (x) = 5×3 +10x +2 = 17+25(x −1)+15(x −1)2 +5(x −1)3.
(Question) Express f (x) = 1 + x + x2 as a linear combination of powers of (x + 2).
Ali Mousavidehshikh
Week 3
Sections 6.5 and 7.1
(Theorem) Let f0(x), f1(x), . . . , fn(x) be non-zero polynomials in Pn. Assume that real numbers a0, a1, . . . , an exists such that
(1) fi(ai) ̸= 0 for each i = 0,1,…,n,
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
(Theorem) Let f0(x), f1(x), . . . , fn(x) be non-zero polynomials in Pn. Assume that real numbers a0, a1, . . . , an exists such that
(1) fi(ai) ̸= 0 for each i = 0,1,…,n, (2) fi(aj) = 0 if i ̸= j.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
(Theorem) Let f0(x), f1(x), . . . , fn(x) be non-zero polynomials in Pn. Assume that real numbers a0, a1, . . . , an exists such that
(1) fi(ai) ̸= 0 for each i = 0,1,…,n,
(2) fi(aj) = 0 if i ̸= j.Then,
1. {f0(x),f1(x),…,fn(x)}isabasisofPn.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
(Theorem) Let f0(x), f1(x), . . . , fn(x) be non-zero polynomials in Pn. Assume that real numbers a0, a1, . . . , an exists such that
(1) fi(ai) ̸= 0 for each i = 0,1,…,n,
(2) fi(aj) = 0 if i ̸= j.Then,
1. {f0(x),f1(x),…,fn(x)}isabasisofPn.
2. If f (x) is any polynomial in Pn, its expansion as a linear
combination of these basis vectors is f(x)=n f(ak)fk(x).
k=0 fk(ak)
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
(Theorem) Let f0(x), f1(x), . . . , fn(x) be non-zero polynomials in Pn. Assume that real numbers a0, a1, . . . , an exists such that
(1) fi(ai) ̸= 0 for each i = 0,1,…,n,
(2) fi(aj) = 0 if i ̸= j.Then,
1. {f0(x),f1(x),…,fn(x)}isabasisofPn.
2. If f (x) is any polynomial in Pn, its expansion as a linear
combination of these basis vectors is f(x)=n f(ak)fk(x).
k=0 fk(ak)
Show that {x2 −x,x2 −2x,x2 −3x +2} is a basis of P2 and express f (x ) = 1 + x as a linear combination of these polynomials.
Ali Mousavidehshikh
Week 3
Sections 6.5 and 7.1
If a0, a1, . . . , an are distinct real numbers, define the Lagrange polynomials δ0(x), δ1(x), . . . , δn(x) relative to these numbers as follows:
i ̸=k (x − ai )
δk (x ) = i ̸=k (ak − ai ) for k = 0, 1, . . . , n.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
If a0, a1, . . . , an are distinct real numbers, define the Lagrange polynomials δ0(x), δ1(x), . . . , δn(x) relative to these numbers as follows:
i ̸=k (x − ai )
δk (x ) = i ̸=k (ak − ai ) for k = 0, 1, . . . , n.
(Example) If n = 3, then
δ1(x) = (x −a0)(x −a2)(x −a3) .
(a1 − a0)(a1 − a2)(a1 − a3)
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
If a0, a1, . . . , an are distinct real numbers, define the Lagrange polynomials δ0(x), δ1(x), . . . , δn(x) relative to these numbers as follows:
i ̸=k (x − ai )
δk (x ) = i ̸=k (ak − ai ) for k = 0, 1, . . . , n.
(Example) If n = 3, then
δ1(x) = (x −a0)(x −a2)(x −a3) .
(a1 − a0)(a1 − a2)(a1 − a3)
Notice that δi(ai) = 1 for each i = 0,1,…,n and δi(aj) = 0 for i ̸= j.
Ali Mousavidehshikh
Week 3
Sections 6.5 and 7.1
If a0, a1, . . . , an are distinct real numbers, define the Lagrange polynomials δ0(x), δ1(x), . . . , δn(x) relative to these numbers as follows:
i ̸=k (x − ai )
δk (x ) = i ̸=k (ak − ai ) for k = 0, 1, . . . , n.
(Example) If n = 3, then
δ1(x) = (x −a0)(x −a2)(x −a3) .
(a1 − a0)(a1 − a2)(a1 − a3)
Notice that δi(ai) = 1 for each i = 0,1,…,n and δi(aj) = 0 for i ̸= j.This gives us the following theorem.
Ali Mousavidehshikh
Week 3
Sections 6.5 and 7.1
(Theorem: Lagrange Interpolation Expansion) Let
a0, a1, . . . , an be distinct real numbers. The corresponding set {δ0(x), δ1(x), . . . , δn(x)} of Lagrange polynomials is a basis for Pn.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
(Theorem: Lagrange Interpolation Expansion) Let
a0, a1, . . . , an be distinct real numbers. The corresponding set {δ0(x), δ1(x), . . . , δn(x)} of Lagrange polynomials is a basis for Pn. Moreover, any polynomial f (x) ∈ Pn has the following unique expression:
n
f (x ) = f (ak )δk (x ).
k=0
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
(Theorem: Lagrange Interpolation Expansion) Let
a0, a1, . . . , an be distinct real numbers. The corresponding set {δ0(x), δ1(x), . . . , δn(x)} of Lagrange polynomials is a basis for Pn. Moreover, any polynomial f (x) ∈ Pn has the following unique expression:
n
f (x ) = f (ak )δk (x ).
k=0
(Question) Find the Lagrange interpolation expansion for f(x)=x2−2x+1relativetoa0 =−1,a1 =0,and
a2 = 1.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
(Theorem: Lagrange Interpolation Expansion) Let
a0, a1, . . . , an be distinct real numbers. The corresponding set {δ0(x), δ1(x), . . . , δn(x)} of Lagrange polynomials is a basis for Pn. Moreover, any polynomial f (x) ∈ Pn has the following unique expression:
n
f (x ) = f (ak )δk (x ).
k=0
(Question) Find the Lagrange interpolation expansion for f(x)=x2−2x+1relativetoa0 =−1,a1 =0,and
a2 = 1. Express f(x) = 2×2 +x +3 as a linear combination of these polynomials.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
(Theorem: Lagrange Interpolation Expansion) Let
a0, a1, . . . , an be distinct real numbers. The corresponding set {δ0(x), δ1(x), . . . , δn(x)} of Lagrange polynomials is a basis for Pn. Moreover, any polynomial f (x) ∈ Pn has the following unique expression:
n
f (x ) = f (ak )δk (x ).
k=0
(Question) Find the Lagrange interpolation expansion for f(x)=x2−2x+1relativetoa0 =−1,a1 =0,and
a2 = 1. Express f(x) = 2×2 +x +3 as a linear combination of these polynomials.
Letf(x)∈Pn,andleta0,a1,…,an denotedistinctreal numbers. If f (ai ) = 0 for all i = 0, 1, . . . , n, then f (x) is the zero polynomial.
Ali Mousavidehshikh
Week 3
Sections 6.5 and 7.1
Definition: If V and W are two vector spaces, a function
T : V → W is called a linear transformation if it satisfies the following two conditions:
(1) T(u + z) = T(u) + T(z) for all u,z ∈ V
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
Definition: If V and W are two vector spaces, a function
T : V → W is called a linear transformation if it satisfies the following two conditions:
(1) T(u + z) = T(u) + T(z) for all u,z ∈ Vand
(2) T(ru) = rT(u) for all r ∈ R and u ∈ V.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
Definition: If V and W are two vector spaces, a function
T : V → W is called a linear transformation if it satisfies the following two conditions:
(1) T(u + z) = T(u) + T(z) for all u,z ∈ Vand
(2) T(ru) = rT(u) for all r ∈ R and u ∈ V.
Notice that the addition inside the brackets is carried out in V while the one on the right is carried out in W . Similar idea for scalar multiplication.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
Definition: If V and W are two vector spaces, a function
T : V → W is called a linear transformation if it satisfies the following two conditions:
(1) T(u + z) = T(u) + T(z) for all u,z ∈ Vand
(2) T(ru) = rT(u) for all r ∈ R and u ∈ V.
Notice that the addition inside the brackets is carried out in V while the one on the right is carried out in W . Similar idea for scalar multiplication.
From MAT223 (Section 2.6, Theorem 2) if T : Rn → Rm is a linear transformation, then writing vectors in Rn as a column matrix, shows that there exists an m × n matrix A such that T(x) = Ax for every x ∈ Rn. In fact,
A = [T(e1) T(e2) ···T(en)], where {e1,e2,…,en} is the standard basis of Rn. We denote this linear transformation by TA :Rn →Rm,definedbyTA(x)=Ax.
Ali Mousavidehshikh
Week 3
Sections 6.5 and 7.1
1V : V → V , the identity linear transformation.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
1V : V → V , the identity linear transformation. 0 : V → W , the zero linear transformation.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
1V : V → V , the identity linear transformation. 0 : V → W , the zero linear transformation.
a : V → V , scalar linear transformation, this one is given by a(v) = av.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
1V : V → V , the identity linear transformation. 0 : V → W , the zero linear transformation.
a : V → V , scalar linear transformation, this one is given by a(v) = av.
ThemapR:Mmn →Mnm givenbyR(A)=AT isalinear transformation.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
1V : V → V , the identity linear transformation. 0 : V → W , the zero linear transformation.
a : V → V , scalar linear transformation, this one is given by a(v) = av.
ThemapR:Mmn →Mnm givenbyR(A)=AT isalinear transformation.
ThemapS:Mnn →RgivenbyS(A)=tr(A)isalinear transformation.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
1V : V → V , the identity linear transformation. 0 : V → W , the zero linear transformation.
a : V → V , scalar linear transformation, this one is given by a(v) = av.
ThemapR:Mmn →Mnm givenbyR(A)=AT isalinear transformation.
ThemapS:Mnn →RgivenbyS(A)=tr(A)isalinear transformation.
Ifa∈R,defineEa :Pn →RviaEa(p(x))=p(a)foreach p(x) ∈ Pn. The map Ea is a linear transformation (this is called evaluation at a).
Ali Mousavidehshikh
Week 3
Sections 6.5 and 7.1
Fixn∈N. ThemapD:Pn →Pn−1 givenby D(p(x)) = p′(x) is a linear transformation.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
Fixn∈N. ThemapD:Pn →Pn−1 givenby D(p(x)) = p′(x) is a linear transformation.
Fixanintegern≥0. ThemapI :Pn →Pn+1 givenby
I(p) = x p(t)dt is a linear transformation. 0
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
Fixn∈N. ThemapD:Pn →Pn−1 givenby D(p(x)) = p′(x) is a linear transformation.
Fixanintegern≥0. ThemapI :Pn →Pn+1 givenby I(p) = x p(t)dt is a linear transformation.
0
Let T : V → W be a linear transformation.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
Fixn∈N. ThemapD:Pn →Pn−1 givenby D(p(x)) = p′(x) is a linear transformation.
Fixanintegern≥0. ThemapI :Pn →Pn+1 givenby I(p) = x p(t)dt is a linear transformation.
0
Let T : V → W be a linear transformation.Then (1) T(0) = 0.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
Fixn∈N. ThemapD:Pn →Pn−1 givenby D(p(x)) = p′(x) is a linear transformation.
Fixanintegern≥0. ThemapI :Pn →Pn+1 givenby I(p) = x p(t)dt is a linear transformation.
0
Let T : V → W be a linear transformation.Then (1) T(0) = 0.
(2) T(−v) = −T(v) for all v ∈ V.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
Fixn∈N. ThemapD:Pn →Pn−1 givenby D(p(x)) = p′(x) is a linear transformation.
Fixanintegern≥0. ThemapI :Pn →Pn+1 givenby I(p) = x p(t)dt is a linear transformation.
0
Let T : V → W be a linear transformation.Then (1) T(0) = 0.
(2) T(−v) = −T(v) for all v ∈ V.
(3) T(ki=1 rivi) = ki=1 riT(vi).
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
Fixn∈N. ThemapD:Pn →Pn−1 givenby D(p(x)) = p′(x) is a linear transformation.
Fixanintegern≥0. ThemapI :Pn →Pn+1 givenby I(p) = x p(t)dt is a linear transformation.
0
Let T : V → W be a linear transformation.Then (1) T(0) = 0.
(2) T(−v) = −T(v) for all v ∈ V.
(3) T(ki=1 rivi) = ki=1 riT(vi).
Let T : V → W be a linear transformation. If T(2v −w) = u and T(v + 3w) = u1, find T(v) and T(w) in terms of u and u1.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
Fixn∈N. ThemapD:Pn →Pn−1 givenby D(p(x)) = p′(x) is a linear transformation.
Fixanintegern≥0. ThemapI :Pn →Pn+1 givenby I(p) = x p(t)dt is a linear transformation.
0
Let T : V → W be a linear transformation.Then (1) T(0) = 0.
(2) T(−v) = −T(v) for all v ∈ V.
(3) T(ki=1 rivi) = ki=1 riT(vi).
Let T : V → W be a linear transformation. If T(2v −w) = u and T(v + 3w) = u1, find T(v) and T(w) in terms of u and u1.
Theorem: Suppose T , S : V → W are two linear transformations, and V = span{v1, v2, . . . , vn}. If T(vi) = S(vi) for each i = 1,2,…,n, then T = S.
Ali Mousavidehshikh
Week 3
Sections 6.5 and 7.1
Theorem: Let V and W be vector spaces and let {u1,u2,…,un} be a basis of V. Given any vectors
z1, z2, . . . zn ∈ W (they need not be distinct), there exists a unique linear transformation T : V → W satisfying
T(ui) = zi for each i = 1,2,…n.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
Theorem: Let V and W be vector spaces and let {u1,u2,…,un} be a basis of V. Given any vectors
z1, z2, . . . zn ∈ W (they need not be distinct), there exists a unique linear transformation T : V → W satisfying
T(ui) = zi for each i = 1,2,…n.In fact, given v = ni=1 aiui (where ai ∈ R), T(v) = ni=1 aizi.
Week 3
Ali Mousavidehshikh
Sections 6.5 and 7.1
Theorem: Let V and W be vector spaces and let {u1,u2,…,un} be a basis of V. Given any vectors
z1, z2, . . . zn ∈ W (they need not be distinct), there exists a unique linear transformation T : V → W satisfying
T(ui) = zi for each i = 1,2,…n.In fact, given v = ni=1 aiui (where ai ∈ R), T(v) = ni=1 aizi.
Find a linear transformation T : P2 → M22 such that 1 0 2
T(1+x)= 0 1 =T(x+x)and
2 11 T(1+x )= 0 0 .
Ali Mousavidehshikh
Week 3
Sections 6.5 and 7.1
Theorem: Let V and W be vector spaces and let {u1,u2,…,un} be a basis of V. Given any vectors
z1, z2, . . . zn ∈ W (they need not be distinct), there exists a unique linear transformation T : V → W satisfying
T(ui) = zi for each i = 1,2,…n.In fact, given v = ni=1 aiui (where ai ∈ R), T(v) = ni=1 aizi.
Find a linear transformation T : P2 → M22 such that 1 0 2
T(1+x)= 0 1 =T(x+x)and
2 11 2
T(1+x)= 0 0 .Noticethatifp(x)=a+bx+cx,then
p(x) = ((a + b − c)(1 + x)/2 + (−a + b + c)(x + x2)/2 + (a − b + c)(1 + x2)/2. Use this to find T(p(x)).
Ali Mousavidehshikh
Week 3