Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

CS 130: Mathematical Methods in Computer Science Vector Spaces and Linear Transformations Nestine Hope S. Hernandez Algorithms and Complexity Laboratory Department of Computer Science University of the Philippines, Diliman nshernandez@dcs.upd.edu.ph Day 7

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Vector Spaces and Linear Transformations

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Vector Spaces and Linear Linear Transformations

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Definition Let V and W be vector spaces. A linear transformation L of V into W is a function assigning a unique vector L(u) in W to each u in V such that

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Definition Let V and W be vector spaces. A linear transformation L of V into W is a function assigning a unique vector L(u) in W to each u in V such that (a) L(u + v) = L(u) + L(v), for every u and v in V . (b) L(ku) = kL(u), for every u in V and every scalar k.

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Definition Let V and W be vector spaces. A linear transformation L of V into W is a function assigning a unique vector L(u) in W to each u in V such that (a) L(u + v) = L(u) + L(v), for every u and v in V . (b) L(ku) = kL(u), for every u in V and every scalar k. If V = W , the linear transformation L : V â†’ V is also called a linear operator on V.

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Definition Let V and W be vector spaces. A linear transformation L of V into W is a function assigning a unique vector L(u) in W to each u in V such that (a) L(u + v) = L(u) + L(v), for every u and v in V . (b) L(ku) = kL(u), for every u in V and every scalar k. If V = W , the linear transformation L : V â†’ V is also called a linear operator on V.

Example 1. L : lR3 â†’ lR2 defined by L(x, y, z) = (x, y)

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Definition Let V and W be vector spaces. A linear transformation L of V into W is a function assigning a unique vector L(u) in W to each u in V such that (a) L(u + v) = L(u) + L(v), for every u and v in V . (b) L(ku) = kL(u), for every u in V and every scalar k. If V = W , the linear transformation L : V â†’ V is also called a linear operator on V.

Example 1. L : lR3 â†’ lR2 defined by L(x, y, z) = (x, y)

(projection)

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Definition Let V and W be vector spaces. A linear transformation L of V into W is a function assigning a unique vector L(u) in W to each u in V such that (a) L(u + v) = L(u) + L(v), for every u and v in V . (b) L(ku) = kL(u), for every u in V and every scalar k. If V = W , the linear transformation L : V → V is also called a linear operator on V.

Example 1. L : lR3 → lR2 defined by L(x, y, z) = (x, y) 3

3

2. L : lR → lR defined by L(u) = ru, r > 1

(projection)

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Definition Let V and W be vector spaces. A linear transformation L of V into W is a function assigning a unique vector L(u) in W to each u in V such that (a) L(u + v) = L(u) + L(v), for every u and v in V . (b) L(ku) = kL(u), for every u in V and every scalar k. If V = W , the linear transformation L : V → V is also called a linear operator on V.

Example 1. L : lR3 → lR2 defined by L(x, y, z) = (x, y) 3

3

2. L : lR → lR defined by L(u) = ru, r > 1

(projection) (dilation)

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Definition Let V and W be vector spaces. A linear transformation L of V into W is a function assigning a unique vector L(u) in W to each u in V such that (a) L(u + v) = L(u) + L(v), for every u and v in V . (b) L(ku) = kL(u), for every u in V and every scalar k. If V = W , the linear transformation L : V → V is also called a linear operator on V.

Example 1. L : lR3 → lR2 defined by L(x, y, z) = (x, y) 3

3

3

3

2. L : lR → lR defined by L(u) = ru, r > 1 3. L : lR → lR defined by L(u) = ru, 0 < r < 1

(projection) (dilation)

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Example 1. L : lR3 → lR2 defined by L(x, y, z) = (x, y)

(projection)

3

3

(dilation)

3

3

(contraction)

2. L : lR → lR defined by L(u) = ru, r > 1 3. L : lR → lR defined by L(u) = ru, 0 < r < 1

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Example 1. L : lR3 → lR2 defined by L(x, y, z) = (x, y)

(projection)

3

3

(dilation)

3

3

(contraction)

2

2

2. L : lR → lR defined by L(u) = ru, r > 1 3. L : lR → lR defined by L(u) = ru, 0 < r < 1 4. L : lR → lR defined by L(x, y) = (x, −y)

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Example 1. L : lR3 → lR2 defined by L(x, y, z) = (x, y)

(projection)

3

3

(dilation)

3

3

(contraction)

2

2

(reflection)

2. L : lR → lR defined by L(u) = ru, r > 1 3. L : lR → lR defined by L(u) = ru, 0 < r < 1 4. L : lR → lR defined by L(x, y) = (x, −y)

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Example 1. L : lR3 → lR2 defined by L(x, y, z) = (x, y)

(projection)

3

3

(dilation)

3

3

(contraction)

2

2

2. L : lR → lR defined by L(u) = ru, r > 1 3. L : lR → lR defined by L(u) = ru, 0 < r < 1

(reflection) 4. L : lR → lR defined by L(x, y) = (x, −y) cos φ − sin φ 5. L : lR2 → lR2 defined by L(u) = u sin φ cos φ

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Example 1. L : lR3 → lR2 defined by L(x, y, z) = (x, y)

(projection)

3

3

(dilation)

3

3

(contraction)

2

2

2. L : lR → lR defined by L(u) = ru, r > 1 3. L : lR → lR defined by L(u) = ru, 0 < r < 1

(reflection) 4. L : lR → lR defined by L(x, y) = (x, −y) cos φ − sin φ 5. L : lR2 → lR2 defined by L(u) = u (rotation) sin φ cos φ

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

More Examples 1. Let L : P1 → P2 be defined by L(at + b) = t(at + b). Show that L is a linear transformation. 2. Let L : P1 → P2 be defined by L[p(t)] = t p(t) + t2 . Is L a linear transformation? 3. Let L : Mmn → Mnm be defined by L(A) = AT for A ∈ Mmn . Is L a linear transformation?

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Some basic properties ... If L : V → W is a linear transformation, then L(c1 v1 + c2 v2 + · · · + ck vk ) = c1 L(v1 ) + c2 L(v2 ) + · · · + ck L(vk ) for any vectors v1 , v2 , · · · , vk in V and any scalars c1 , c2 , · · · , ck .

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Some basic properties ... If L : V → W is a linear transformation, then L(c1 v1 + c2 v2 + · · · + ck vk ) = c1 L(v1 ) + c2 L(v2 ) + · · · + ck L(vk ) for any vectors v1 , v2 , · · · , vk in V and any scalars c1 , c2 , · · · , ck . Let L : V → W be a linear transformation. Then (a) L(0V ) = 0W , (b) L(u − v) = L(u) − L(v).

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Some basic properties ... If L : V → W is a linear transformation, then L(c1 v1 + c2 v2 + · · · + ck vk ) = c1 L(v1 ) + c2 L(v2 ) + · · · + ck L(vk ) for any vectors v1 , v2 , · · · , vk in V and any scalars c1 , c2 , · · · , ck . Let L : V → W be a linear transformation. Then (a) L(0V ) = 0W , (b) L(u − v) = L(u) − L(v).

Let T : V → W be a function. If T (0V ) 6= 0W , then T is not a linear transformation.

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Some basic properties ... If L : V → W is a linear transformation, then L(c1 v1 + c2 v2 + · · · + ck vk ) = c1 L(v1 ) + c2 L(v2 ) + · · · + ck L(vk ) for any vectors v1 , v2 , · · · , vk in V and any scalars c1 , c2 , · · · , ck . Let L : V → W be a linear transformation. Then (a) L(0V ) = 0W , (b) L(u − v) = L(u) − L(v).

Let T : V → W be a function. If T (0V ) 6= 0W , then T is not a linear transformation. Let L : V → W be a linear transformation of an n-dimensional vector space V into a vector space W . Also, let S = {v1 , v2 , · · · , vn } be a basis for V . If u is any vector in V , then L(u) is completely determined by {L(v1 ), L(v2 ), · · · , L(vn )}.

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Vector Spaces and Linear Linear Transformations

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

One-to-One Linear Transformations

A linear transformation L : V â†’ W is said to be one-to-one if for all v1 , v2 in V , v1 6= v2 implies that L(v1 ) 6= L(v2 ).

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

One-to-One Linear Transformations

A linear transformation L : V â†’ W is said to be one-to-one if for all v1 , v2 in V , v1 6= v2 implies that L(v1 ) 6= L(v2 ). An equivalent statement is that L is one-to-one if for all v1 , v2 in V , L(v1 ) = L(v2 ) implies that v1 = v2 .

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Kernel Let L : V â†’ W be a linear transformation. Then kernel of L, ker L, is the subset of V consisting of all vectors v such that L(v) = 0W .

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Kernel Let L : V → W be a linear transformation. Then kernel of L, ker L, is the subset of V consisting of all vectors v such that L(v) = 0W .

Example 02

31 x » – 6 B 7C 6 y 7C = x + y , If L : lR4 → lR2 is defined by L B @4 z 5A z+w w then the ker L consists of

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Kernel Let L : V → W be a linear transformation. Then kernel of L, ker L, is the subset of V consisting of all vectors v such that L(v) = 0W .

Example 02

31 x » – 6 B 7C 6 y 7C = x + y , If L : lR4 → lR2 is defined by L B @4 z 5A z+w w 2 3 r 6 −r 7 7 then the ker L consists of all vectors of the form 6 4 s 5 ∀ r, s ∈ lR. −s

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Kernel Let L : V → W be a linear transformation. Then kernel of L, ker L, is the subset of V consisting of all vectors v such that L(v) = 0W .

Example 02

31 x » – 6 B 7C 6 y 7C = x + y , If L : lR4 → lR2 is defined by L B @4 z 5A z+w w 2 3 r 6 −r 7 7 then the ker L consists of all vectors of the form 6 4 s 5 ∀ r, s ∈ lR. −s

If L : V → W is a linear transformation, then ker L is a subspace of V.

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Example 02

31 x » – 6 B 7C 6 y 7C = x + y , If L : lR4 → lR2 is defined by L B @4 z 5A z+w w 2 3 r 6 −r 7 7 then the ker L consists of all vectors of the form 6 4 s 5 ∀ r, s ∈ lR. −s

If L : V → W is a linear transformation, then ker L is a subspace of V. A linear transformation L : V → W is one-to-one if and only if ker L = {0V }.

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Example 02

If L : V → W is a linear transformation, then ker L is a subspace of V. A linear transformation L : V → W is one-to-one if and only if ker L = {0V }. Note: L is one-to-one if and only if dim(ker L) = 0.

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Range

If L : V â†’ W is a linear transformation, then the range of L, denoted by range L, is the set of all vectors in W that are images, under L, of vectors in V .

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Range

If L : V â†’ W is a linear transformation, then the range of L, denoted by range L, is the set of all vectors in W that are images, under L, of vectors in V . Thus a vector w is in the range of L if there exists some vector v in V such that L(v) = w.

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Range

If L : V â†’ W is a linear transformation, then the range of L, denoted by range L, is the set of all vectors in W that are images, under L, of vectors in V . Thus a vector w is in the range of L if there exists some vector v in V such that L(v) = w.

If range L = W , we say that L is onto.

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Range

If L : V â†’ W is a linear transformation, then the range of L, denoted by range L, is the set of all vectors in W that are images, under L, of vectors in V . Thus a vector w is in the range of L if there exists some vector v in V such that L(v) = w.

If range L = W , we say that L is onto.

If L : V â†’ W is a linear transformation, then range L is a subspace of W .

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

An Example

Let lR3 → lR3 be defined by a1 1 L a2 = 1 a3 2

0 1 1

(a) Is L onto? (b) Find a basis for range L. (c) Find ker L. (d) Is L one-to-one?

1 a1 2 a2 . 3 a3

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Rank and Nullity

If L : V â†’ W is a linear transformation of an nâˆ’dimensional space vector space V into a vector space W , then dim(range L) + dim(ker L) = dim(V ).

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Rank and Nullity

If L : V â†’ W is a linear transformation of an nâˆ’dimensional space vector space V into a vector space W , then dim(range L) + dim(ker L) = dim(V ).

The dimension of range L is called the rank of L, and the dimension of ker L is also called the nullity of L.

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

List of Nonsingular Equivalences Suppose A is an n × n matrix. Then the following statements are equivalent: 1. A is nonsingular. 2. A is row equivalent to In . 3. Ax = 0 has only the trivial solution. 4. The linear system Ax = b has a unique solution for every n × 1 matrix b. 5. det(A) 6= 0. 6. A has rank n. 7. A has nullity 0. 8. The rows of A form a linearly independent set of n vectors in lRn . 9. The columns of A form a linearly independent set of n vectors in lRn . 10. Zero is NOT an eigenvalue of A. 11. The linear operator L : lRn → lRn defined by L(x) = A(x), for x in lRn , is one-to-one and onto.

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Vector Spaces and Linear Linear Transformations

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Matrix of a Linear Transformation Let L : lRn → lRm be a linear transformation. Then there exists a unique m × n matrix A such that L(x) = Ax for x ∈ lRn . Note: consider vectors in lRn and lRm as n × 1 and m × 1 matrices, respectively.

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Matrix Representation of a Linear Transformation with respect to given bases

Let L : V → W be a linear transformation of an n−dimensional vector space V into an m−dimensional vector space W and let S = {v1 , v2 , · · · , vn } and T = {w1 , w2 , · · · , wm } be bases for V and W , respectively.

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Matrix Representation of a Linear Transformation with respect to given bases

Let L : V → W be a linear transformation of an n−dimensional vector space V into an m−dimensional vector space W and let S = {v1 , v2 , · · · , vn } and T = {w1 , w2 , · · · , wm } be bases for V and W , respectively. Then the m × n matrix A, whose jth column is the coordinate vector [L(vj )]T of L(vj ) with respect to T , is associated with L and has the following property:

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Matrix Representation of a Linear Transformation with respect to given bases

Let L : V → W be a linear transformation of an n−dimensional vector space V into an m−dimensional vector space W and let S = {v1 , v2 , · · · , vn } and T = {w1 , w2 , · · · , wm } be bases for V and W , respectively. Then the m × n matrix A, whose jth column is the coordinate vector [L(vj )]T of L(vj ) with respect to T , is associated with L and has the following property: If x is in V , then [L(x)]T = A[x]S .

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Matrix Representation of a Linear Transformation with respect to given bases

Let L : V → W be a linear transformation of an n−dimensional vector space V into an m−dimensional vector space W and let S = {v1 , v2 , · · · , vn } and T = {w1 , w2 , · · · , wm } be bases for V and W , respectively. Then the m × n matrix A, whose jth column is the coordinate vector [L(vj )]T of L(vj ) with respect to T , is associated with L and has the following property: If x is in V , then [L(x)]T = A[x]S . Moreover, A is the only matrix with this property.

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Matrix Representation of a Linear Transformation with respect to given bases

Let L : V → W be a linear transformation of an n−dimensional vector space V into an m−dimensional vector space W and let S = {v1 , v2 , · · · , vn } and T = {w1 , w2 , · · · , wm } be bases for V and W , respectively. Then the m × n matrix A, whose jth column is the coordinate vector [L(vj )]T of L(vj ) with respect to T , is associated with L and has the following property: If x is in V , then [L(x)]T = A[x]S . Moreover, A is the only matrix with this property.

Example:

Find the matrix of L with respect to S and T .

x 1 3 2 y = Let L : lR → lR be defined by L 1 z

1 2

1 3

x y . z

• Let S and T be the natural bases for lR3 and lR2 respectively.

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Matrix Representation of a Linear Transformation with respect to given bases

Let L : V â†’ W be a linear transformation of an nâˆ’dimensional vector space V into an mâˆ’dimensional vector space W and let S = {v1 , v2 , Âˇ Âˇ Âˇ , vn } and T = {w1 , w2 , Âˇ Âˇ Âˇ , wm } be bases for V and W , respectively. Then the m Ă— n matrix A, whose jth column is the coordinate vector [L(vj )]T of L(vj ) with respect to T , is associated with L and has the following property: If x is in V , then [L(x)]T = A[x]S . Moreover, A is the only matrix with this property.

Example:

Find the matrix of L with ďŁŤďŁŽrespect ďŁšďŁśto S and T .

x 1 3 2 ďŁ ďŁ° ďŁť ďŁ¸ y = Let L : lR â†’ lR be defined by L 1 z

1 2

1 3

ďŁŽ

ďŁš x ďŁ° y ďŁť. z

â€˘ Let S and T be the natural bases for lR3 and lR2 respectively.

ďŁąďŁŽ ďŁš ďŁŽ ďŁš ďŁŽ ďŁšďŁź 0 0 ďŁ˝ ďŁ˛ 1 1 1 â€˘ Let S = ďŁ° 1 ďŁť , ďŁ° 1 ďŁť , ďŁ° 0 ďŁť and T = , 2 3 ďŁł ďŁž 0 1 1

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

More Examples Let L : P1 → P2 be defined by L[p(t)] = t p(t). (a) Find the matrix of L with respect to the bases S = {t, 1} and T = {t2 , t, 1} for P1 and P2 , respectively. (b) If p(t) = 3t − 2, compute L[p(t)] directly and using the matrix obtained in (a). (c) Find the matrix of L with respect to the bases S = {t, 1} and T = {t2 , t − 1, t + 1} for P1 and P2 , respectively. (d) If p(t) = 3t − 2, compute L[p(t)] using the matrix obtained in (c).

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Matrix Representation of a Linear Operator

Theorem Let L : V → V be a linear operator, where V is an n-dimensional vector space. Let S = {v1 , v2 , · · · , vn } and T = {w1 , w2 , · · · , wn } be bases for V and let PS←T be the transition matrix from T to S. If A is the matrix representing L with respect to S, then P −1 AP is the matrix representing L with respect to the basis T .

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Matrix Representation of a Linear Operator

Theorem Let L : V → V be a linear operator, where V is an n-dimensional vector space. Let S = {v1 , v2 , · · · , vn } and T = {w1 , w2 , · · · , wn } be bases for V and let PS←T be the transition matrix from T to S. If A is the matrix representing L with respect to S, then P −1 AP is the matrix representing L with respect to the basis T . x

L

−−−−→

L(x)

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Matrix Representation of a Linear Operator

Theorem Let L : V → V be a linear operator, where V is an n-dimensional vector space. Let S = {v1 , v2 , · · · , vn } and T = {w1 , w2 , · · · , wn } be bases for V and let PS←T be the transition matrix from T to S. If A is the matrix representing L with respect to S, then P −1 AP is the matrix representing L with respect to the basis T . x

L

−−−−→

y x T

L(x) y

B

−−−−−→

L(x) T = B x T

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Matrix Representation of a Linear Operator

Theorem Let L : V → V be a linear operator, where V is an n-dimensional vector space. Let S = {v1 , v2 , · · · , vn } and T = {w1 , w2 , · · · , wn } be bases for V and let PS←T be the transition matrix from T to S. If A is the matrix representing L with respect to S, then P −1 AP is the matrix representing L with respect to the basis T . x

L

−−−−→

y x T

y B

−−−−−→

y

L(x) T = B x T = P −1 AP x T x -1 P

P

x S =P x T

L(x)

A

−−−−−→

AP x T = L(x) S

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Example Find the matrix of L with respect to S and T .

Let L : lR2 â†’ lR2 be defined by L Let S =

1 0

a1 a2

=

a1 + a2 a1 âˆ’ 2a2

0 1 2 , and T = , . 1 âˆ’1 1

.

Linear Transformation

Kernel and Range

Questions???

The Matrix of a Linear Transformation

Linear Transformation

Kernel and Range

The Matrix of a Linear Transformation

Questions??? Reminder : Next meeting is your FIRST LONG EXAM! Study hard!!! See you!