Dimension Subspace of a Finite-Dimensional Space

Sec 4.5 The Dimension of a Vector Space

Xiaozhou Li

xiaozhouli@uestc.edu.cn

School of Mathematical Sciences,UESTC, Chengdu, China

Dimension Subspace of a Finite-Dimensional Space

Outline

1 Dimension

2 Subspace of a Finite-Dimensional Space

Dimension Subspace of a Finite-Dimensional Space

Outline

1 Dimension

2 Subspace of a Finite-Dimensional Space

Dimension Subspace of a Finite-Dimensional Space

The coordinate mapping shows that a vector with a basiscontaining n vectors is isomorphic to Rn.

Theorem

If a vector space V has a basis B = {b1, . . . , bn}, then any setin V containing more than n vectors must be linearlydependent.

If a vector space V has a basis of n vectors, then every basisof V must consist of exactly n vectors.

Dimension Subspace of a Finite-Dimensional Space

The coordinate mapping shows that a vector with a basiscontaining n vectors is isomorphic to Rn.

Theorem

If a vector space V has a basis B = {b1, . . . , bn}, then any setin V containing more than n vectors must be linearlydependent.

If a vector space V has a basis of n vectors, then every basisof V must consist of exactly n vectors.

Dimension Subspace of a Finite-Dimensional Space

The coordinate mapping shows that a vector with a basiscontaining n vectors is isomorphic to Rn.

Theorem

If a vector space V has a basis B = {b1, . . . , bn}, then any setin V containing more than n vectors must be linearlydependent.

If a vector space V has a basis of n vectors, then every basisof V must consist of exactly n vectors.

Dimension Subspace of a Finite-Dimensional Space

Dimension

DefinitionIf V is spanned by a finite set, then V is said to befinite-dimensional, and the dimension of V , written asdimV , is the number of vector in a basis for V .

The dimension of the zero vector space {0} is defined to bezero.

If V is not spanned by a finite set, then V is said to beinfinite-dimensional.

Dimension Subspace of a Finite-Dimensional Space

Dimension

DefinitionIf V is spanned by a finite set, then V is said to befinite-dimensional, and the dimension of V , written asdimV , is the number of vector in a basis for V .

The dimension of the zero vector space {0} is defined to bezero.

If V is not spanned by a finite set, then V is said to beinfinite-dimensional.

Dimension Subspace of a Finite-Dimensional Space

Dimension

DefinitionIf V is spanned by a finite set, then V is said to befinite-dimensional, and the dimension of V , written asdimV , is the number of vector in a basis for V .

The dimension of the zero vector space {0} is defined to bezero.

If V is not spanned by a finite set, then V is said to beinfinite-dimensional.

Dimension Subspace of a Finite-Dimensional Space

Example

The standard basis for Rn contains n vectors, so dimRn = n.

The standard polynomial basis {1, t, t2} shows that dimP2 = 3.In general, dimPn = n+ 1.

The space P of all polynomials is infinite-dimensional.

Dimension Subspace of a Finite-Dimensional Space

Example

The standard basis for Rn contains n vectors, so dimRn = n.

The standard polynomial basis {1, t, t2} shows that dimP2 = 3.

In general, dimPn = n+ 1.

The space P of all polynomials is infinite-dimensional.

Dimension Subspace of a Finite-Dimensional Space

Example

The standard basis for Rn contains n vectors, so dimRn = n.

The standard polynomial basis {1, t, t2} shows that dimP2 = 3.In general, dimPn = n+ 1.

The space P of all polynomials is infinite-dimensional.

Dimension Subspace of a Finite-Dimensional Space

Example

The standard basis for Rn contains n vectors, so dimRn = n.

The standard polynomial basis {1, t, t2} shows that dimP2 = 3.In general, dimPn = n+ 1.

The space P of all polynomials is infinite-dimensional.

Dimension Subspace of a Finite-Dimensional Space

Example

Find the dimension of the subspace

H =

a− 3b+ 6c5a+ 4d

b− 2c− d5d

: a, b, c, d ∈ R

Dimension Subspace of a Finite-Dimensional Space

Example

The subspaces of R3:

0-dimensional subspaces.

7KH 'LPHQVLRQ RI D 9HFWRU 6SDFH

��GLPHQVLRQDO� VXEVSDFHV� $Q\� VXEVSDFH� VSDQQHG� E\� WZR� OLQHDUO\� LQGHSHQGHQWYHFWRUV� 6XFK�VXEVSDFHV�DUH�SODQHV�WKURXJK�WKH�RULJLQ���GLPHQVLRQDO�VXEVSDFHV� 2QO\ R3 LWVHOI� $Q\�WKUHH�OLQHDUO\�LQGHSHQGHQW�YHFWRUVLQ R3 VSDQ�DOO�RI R3� E\�WKH�,QYHUWLEOH�0DWUL[�7KHRUHP�

x2

x1

(a)

0-dim x2

x1

(b) (c)

1-dim

3-dim

2-dim2-dim

x1

x2

x3x3x3

6DPSOH�VXEVSDFHV�RI R3�

6XEVSDFHV RI D )LQLWH�'LPHQVLRQDO 6SDFH7KH�QH[W�WKHRUHP�LV�D�QDWXUDO�FRXQWHUSDUW�WR�WKH�6SDQQLQJ�6HW�7KHRUHP�

/HW H EH� D� VXEVSDFH� RI� D� ÀQLWH�GLPHQVLRQDO� YHFWRU� VSDFH V � $Q\� OLQHDUO\LQGHSHQGHQW�VHW�LQ H FDQ�EH�H[SDQGHG� LI�QHFHVVDU\� WR�D�EDVLV�IRU H � $OVR� H LVÀQLWH�GLPHQVLRQDO�DQG

GLPH ! GLPV

,I H D f�g� WKHQ�FHUWDLQO\ GLPH D 0 ! GLPV � 2WKHUZLVH� OHW S D fX1; : : : ;Xkg EH� DQ\� OLQHDUO\� LQGHSHQGHQW� VHW� LQ H � ,I S VSDQV H � WKHQ S LV� D� EDVLV� IRU H �2WKHUZLVH� WKHUH�LV�VRPH XkC1 LQ H WKDW�LV�QRW�LQ 6SDQS � %XW�WKHQ fX1; : : : ; Xk; XkC1gZLOO�EH�OLQHDUO\�LQGHSHQGHQW� EHFDXVH�QR�YHFWRU�LQ�WKH�VHW�FDQ�EH�D�OLQHDU�FRPELQDWLRQ�RIYHFWRUV�WKDW�SUHFHGH�LW��E\�7KHRUHP ���

6R�ORQJ�DV�WKH�QHZ�VHW�GRHV�QRW�VSDQ H � ZH�FDQ�FRQWLQXH�WKLV�SURFHVV�RI�H[SDQGLQJS WR�D� ODUJHU� OLQHDUO\� LQGHSHQGHQW� VHW� LQ H � %XW� WKH�QXPEHU�RI�YHFWRUV� LQ�D� OLQHDUO\LQGHSHQGHQW� H[SDQVLRQ� RI S FDQ� QHYHU� H[FHHG� WKH� GLPHQVLRQ� RI V � E\� 7KHRUHP ��6R�HYHQWXDOO\�WKH�H[SDQVLRQ�RI S ZLOO�VSDQ H DQG�KHQFH�ZLOO�EH�D�EDVLV�IRU H � DQGGLPH ! GLPV �

:KHQ�WKH�GLPHQVLRQ�RI�D�YHFWRU�VSDFH�RU�VXEVSDFH�LV�NQRZQ� WKH�VHDUFK�IRU�D�EDVLVLV�VLPSOLÀHG�E\�WKH�QH[W�WKHRUHP� ,W�VD\V�WKDW�LI�D�VHW�KDV�WKH�ULJKW�QXPEHU�RI�HOHPHQWV�WKHQ�RQH�KDV�RQO\�WR�VKRZ�HLWKHU�WKDW�WKH�VHW�LV�OLQHDUO\�LQGHSHQGHQW�RU�WKDW�LW�VSDQV�WKHVSDFH� 7KH�WKHRUHP�LV�RI�FULWLFDO�LPSRUWDQFH�LQ�QXPHURXV�DSSOLHG�SUREOHPV��LQYROYLQJGLIIHUHQWLDO�HTXDWLRQV�RU�GLIIHUHQFH�HTXDWLRQV� IRU�H[DPSOH��ZKHUH�OLQHDU�LQGHSHQGHQFHLV�PXFK�HDVLHU�WR�YHULI\�WKDQ�VSDQQLQJ�

/HW V EH�D p�GLPHQVLRQDO�YHFWRU�VSDFH� p " 1� $Q\�OLQHDUO\�LQGHSHQGHQW�VHW�RIH[DFWO\ p HOHPHQWV� LQ V LV� DXWRPDWLFDOO\� D� EDVLV� IRU V � $Q\� VHW� RI� H[DFWO\ pHOHPHQWV�WKDW�VSDQV V LV�DXWRPDWLFDOO\�D�EDVLV�IRU V �

6(&21' 5(9,6(' 3$*(6

Dimension Subspace of a Finite-Dimensional Space

Example

The subspaces of R3:

0-dimensional subspaces.

7KH 'LPHQVLRQ RI D 9HFWRU 6SDFH

��GLPHQVLRQDO� VXEVSDFHV� $Q\� VXEVSDFH� VSDQQHG� E\� WZR� OLQHDUO\� LQGHSHQGHQWYHFWRUV� 6XFK�VXEVSDFHV�DUH�SODQHV�WKURXJK�WKH�RULJLQ���GLPHQVLRQDO�VXEVSDFHV� 2QO\ R3 LWVHOI� $Q\�WKUHH�OLQHDUO\�LQGHSHQGHQW�YHFWRUVLQ R3 VSDQ�DOO�RI R3� E\�WKH�,QYHUWLEOH�0DWUL[�7KHRUHP�

x2

x1

(a)

0-dim x2

x1

(b) (c)

1-dim

3-dim

2-dim2-dim

x1

x2

x3x3x3

6DPSOH�VXEVSDFHV�RI R3�

6XEVSDFHV RI D )LQLWH�'LPHQVLRQDO 6SDFH7KH�QH[W�WKHRUHP�LV�D�QDWXUDO�FRXQWHUSDUW�WR�WKH�6SDQQLQJ�6HW�7KHRUHP�

/HW H EH� D� VXEVSDFH� RI� D� ÀQLWH�GLPHQVLRQDO� YHFWRU� VSDFH V � $Q\� OLQHDUO\LQGHSHQGHQW�VHW�LQ H FDQ�EH�H[SDQGHG� LI�QHFHVVDU\� WR�D�EDVLV�IRU H � $OVR� H LVÀQLWH�GLPHQVLRQDO�DQG

GLPH ! GLPV

,I H D f�g� WKHQ�FHUWDLQO\ GLPH D 0 ! GLPV � 2WKHUZLVH� OHW S D fX1; : : : ;Xkg EH� DQ\� OLQHDUO\� LQGHSHQGHQW� VHW� LQ H � ,I S VSDQV H � WKHQ S LV� D� EDVLV� IRU H �2WKHUZLVH� WKHUH�LV�VRPH XkC1 LQ H WKDW�LV�QRW�LQ 6SDQS � %XW�WKHQ fX1; : : : ; Xk; XkC1gZLOO�EH�OLQHDUO\�LQGHSHQGHQW� EHFDXVH�QR�YHFWRU�LQ�WKH�VHW�FDQ�EH�D�OLQHDU�FRPELQDWLRQ�RIYHFWRUV�WKDW�SUHFHGH�LW��E\�7KHRUHP ���

6R�ORQJ�DV�WKH�QHZ�VHW�GRHV�QRW�VSDQ H � ZH�FDQ�FRQWLQXH�WKLV�SURFHVV�RI�H[SDQGLQJS WR�D� ODUJHU� OLQHDUO\� LQGHSHQGHQW� VHW� LQ H � %XW� WKH�QXPEHU�RI�YHFWRUV� LQ�D� OLQHDUO\LQGHSHQGHQW� H[SDQVLRQ� RI S FDQ� QHYHU� H[FHHG� WKH� GLPHQVLRQ� RI V � E\� 7KHRUHP ��6R�HYHQWXDOO\�WKH�H[SDQVLRQ�RI S ZLOO�VSDQ H DQG�KHQFH�ZLOO�EH�D�EDVLV�IRU H � DQGGLPH ! GLPV �

:KHQ�WKH�GLPHQVLRQ�RI�D�YHFWRU�VSDFH�RU�VXEVSDFH�LV�NQRZQ� WKH�VHDUFK�IRU�D�EDVLVLV�VLPSOLÀHG�E\�WKH�QH[W�WKHRUHP� ,W�VD\V�WKDW�LI�D�VHW�KDV�WKH�ULJKW�QXPEHU�RI�HOHPHQWV�WKHQ�RQH�KDV�RQO\�WR�VKRZ�HLWKHU�WKDW�WKH�VHW�LV�OLQHDUO\�LQGHSHQGHQW�RU�WKDW�LW�VSDQV�WKHVSDFH� 7KH�WKHRUHP�LV�RI�FULWLFDO�LPSRUWDQFH�LQ�QXPHURXV�DSSOLHG�SUREOHPV��LQYROYLQJGLIIHUHQWLDO�HTXDWLRQV�RU�GLIIHUHQFH�HTXDWLRQV� IRU�H[DPSOH��ZKHUH�OLQHDU�LQGHSHQGHQFHLV�PXFK�HDVLHU�WR�YHULI\�WKDQ�VSDQQLQJ�

/HW V EH�D p�GLPHQVLRQDO�YHFWRU�VSDFH� p " 1� $Q\�OLQHDUO\�LQGHSHQGHQW�VHW�RIH[DFWO\ p HOHPHQWV� LQ V LV� DXWRPDWLFDOO\� D� EDVLV� IRU V � $Q\� VHW� RI� H[DFWO\ pHOHPHQWV�WKDW�VSDQV V LV�DXWRPDWLFDOO\�D�EDVLV�IRU V �

6(&21' 5(9,6(' 3$*(6

Dimension Subspace of a Finite-Dimensional Space

Example

The subspaces of R3:

0-dimensional subspaces.

7KH 'LPHQVLRQ RI D 9HFWRU 6SDFH

��GLPHQVLRQDO� VXEVSDFHV� $Q\� VXEVSDFH� VSDQQHG� E\� WZR� OLQHDUO\� LQGHSHQGHQWYHFWRUV� 6XFK�VXEVSDFHV�DUH�SODQHV�WKURXJK�WKH�RULJLQ���GLPHQVLRQDO�VXEVSDFHV� 2QO\ R3 LWVHOI� $Q\�WKUHH�OLQHDUO\�LQGHSHQGHQW�YHFWRUVLQ R3 VSDQ�DOO�RI R3� E\�WKH�,QYHUWLEOH�0DWUL[�7KHRUHP�

x2

x1

(a)

0-dim x2

x1

(b) (c)

1-dim

3-dim

2-dim2-dim

x1

x2

x3x3x3

6DPSOH�VXEVSDFHV�RI R3�

6XEVSDFHV RI D )LQLWH�'LPHQVLRQDO 6SDFH7KH�QH[W�WKHRUHP�LV�D�QDWXUDO�FRXQWHUSDUW�WR�WKH�6SDQQLQJ�6HW�7KHRUHP�

/HW H EH� D� VXEVSDFH� RI� D� ÀQLWH�GLPHQVLRQDO� YHFWRU� VSDFH V � $Q\� OLQHDUO\LQGHSHQGHQW�VHW�LQ H FDQ�EH�H[SDQGHG� LI�QHFHVVDU\� WR�D�EDVLV�IRU H � $OVR� H LVÀQLWH�GLPHQVLRQDO�DQG

GLPH ! GLPV

,I H D f�g� WKHQ�FHUWDLQO\ GLPH D 0 ! GLPV � 2WKHUZLVH� OHW S D fX1; : : : ;Xkg EH� DQ\� OLQHDUO\� LQGHSHQGHQW� VHW� LQ H � ,I S VSDQV H � WKHQ S LV� D� EDVLV� IRU H �2WKHUZLVH� WKHUH�LV�VRPH XkC1 LQ H WKDW�LV�QRW�LQ 6SDQS � %XW�WKHQ fX1; : : : ; Xk; XkC1gZLOO�EH�OLQHDUO\�LQGHSHQGHQW� EHFDXVH�QR�YHFWRU�LQ�WKH�VHW�FDQ�EH�D�OLQHDU�FRPELQDWLRQ�RIYHFWRUV�WKDW�SUHFHGH�LW��E\�7KHRUHP ���

6R�ORQJ�DV�WKH�QHZ�VHW�GRHV�QRW�VSDQ H � ZH�FDQ�FRQWLQXH�WKLV�SURFHVV�RI�H[SDQGLQJS WR�D� ODUJHU� OLQHDUO\� LQGHSHQGHQW� VHW� LQ H � %XW� WKH�QXPEHU�RI�YHFWRUV� LQ�D� OLQHDUO\LQGHSHQGHQW� H[SDQVLRQ� RI S FDQ� QHYHU� H[FHHG� WKH� GLPHQVLRQ� RI V � E\� 7KHRUHP ��6R�HYHQWXDOO\�WKH�H[SDQVLRQ�RI S ZLOO�VSDQ H DQG�KHQFH�ZLOO�EH�D�EDVLV�IRU H � DQGGLPH ! GLPV �

:KHQ�WKH�GLPHQVLRQ�RI�D�YHFWRU�VSDFH�RU�VXEVSDFH�LV�NQRZQ� WKH�VHDUFK�IRU�D�EDVLVLV�VLPSOLÀHG�E\�WKH�QH[W�WKHRUHP� ,W�VD\V�WKDW�LI�D�VHW�KDV�WKH�ULJKW�QXPEHU�RI�HOHPHQWV�WKHQ�RQH�KDV�RQO\�WR�VKRZ�HLWKHU�WKDW�WKH�VHW�LV�OLQHDUO\�LQGHSHQGHQW�RU�WKDW�LW�VSDQV�WKHVSDFH� 7KH�WKHRUHP�LV�RI�FULWLFDO�LPSRUWDQFH�LQ�QXPHURXV�DSSOLHG�SUREOHPV��LQYROYLQJGLIIHUHQWLDO�HTXDWLRQV�RU�GLIIHUHQFH�HTXDWLRQV� IRU�H[DPSOH��ZKHUH�OLQHDU�LQGHSHQGHQFHLV�PXFK�HDVLHU�WR�YHULI\�WKDQ�VSDQQLQJ�

/HW V EH�D p�GLPHQVLRQDO�YHFWRU�VSDFH� p " 1� $Q\�OLQHDUO\�LQGHSHQGHQW�VHW�RIH[DFWO\ p HOHPHQWV� LQ V LV� DXWRPDWLFDOO\� D� EDVLV� IRU V � $Q\� VHW� RI� H[DFWO\ pHOHPHQWV�WKDW�VSDQV V LV�DXWRPDWLFDOO\�D�EDVLV�IRU V �

6(&21' 5(9,6(' 3$*(6

Dimension Subspace of a Finite-Dimensional Space

Outline

1 Dimension

2 Subspace of a Finite-Dimensional Space

Dimension Subspace of a Finite-Dimensional Space

TheoremLet H be a subspace of a finite-dimensional vector space V .

Anylinearly independent set H can be expanded, if necessary, to a basisfor H. Also, H is finite dimensional and

dimH ≤ dimV

Dimension Subspace of a Finite-Dimensional Space

TheoremLet H be a subspace of a finite-dimensional vector space V . Anylinearly independent set H can be expanded, if necessary, to a basisfor H.

Also, H is finite dimensional and

dimH ≤ dimV

Dimension Subspace of a Finite-Dimensional Space

TheoremLet H be a subspace of a finite-dimensional vector space V . Anylinearly independent set H can be expanded, if necessary, to a basisfor H. Also, H is finite dimensional and

dimH ≤ dimV

Dimension Subspace of a Finite-Dimensional Space

TheoremLet V be a p-dimensional vector space, p ≥ 1.

Any linearly independent set of exactly p elements in V isautomatically a basis for V .

Any set of exactly p elements that spans V is automatically abasis for V .

Dimension Subspace of a Finite-Dimensional Space

TheoremLet V be a p-dimensional vector space, p ≥ 1.

Any linearly independent set of exactly p elements in V isautomatically a basis for V .

Any set of exactly p elements that spans V is automatically abasis for V .

Dimension Subspace of a Finite-Dimensional Space

TheoremLet V be a p-dimensional vector space, p ≥ 1.

Any linearly independent set of exactly p elements in V isautomatically a basis for V .

Any set of exactly p elements that spans V is automatically abasis for V .

Dimension Subspace of a Finite-Dimensional Space

The Dimension of NulA and ColA

The dimension of NulA is the number of free variables in theequation Ax = 0.

The dimension of ColA is the number of pivot columns in A.

Dimension Subspace of a Finite-Dimensional Space

The Dimension of NulA and ColA

The dimension of NulA is the number of free variables in theequation Ax = 0.

The dimension of ColA is the number of pivot columns in A.

Dimension Subspace of a Finite-Dimensional Space

The Dimension of NulA and ColA

The dimension of NulA is the number of free variables in theequation Ax = 0.

The dimension of ColA is the number of pivot columns in A.

Dimension Subspace of a Finite-Dimensional Space

Example

Find the dimensions of the null space and the column space of

A =

−3 6 −1 1 −71 −2 2 3 −12 −4 5 8 −4

∼

1 −2 2 3 −10 0 1 2 −20 0 0 0 0

dimNulA=3, and dimColA=2.

Dimension Subspace of a Finite-Dimensional Space

Example

Find the dimensions of the null space and the column space of

A =

−3 6 −1 1 −71 −2 2 3 −12 −4 5 8 −4

∼1 −2 2 3 −10 0 1 2 −20 0 0 0 0

dimNulA=3, and dimColA=2.

Dimension Subspace of a Finite-Dimensional Space

Example

Find the dimensions of the null space and the column space of

A =

−3 6 −1 1 −71 −2 2 3 −12 −4 5 8 −4

∼1 −2 2 3 −10 0 1 2 −20 0 0 0 0

dimNulA=3, and dimColA=2.