MA106 Linear Algebra Revision Guide

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MA106

Linear Algebra

Revision GuideWritten by Shriti Somaiya and Jonathan Elliott iiMA106 Linear AlgebraContents

1 Introduction1

2 Vector Spaces2

3 Linear Independence, Spanning Sets and Bases 3

4 Matrices and Linear Maps 5

5 Elementary Operations and the Rank of a Matrix 7

6 Determinants and Inverses 8

7 Change of Basis and Similar Matrices 10

Introduction

This revision guide for MA106 Linear Algebra has been designed as an aid to revision, not a substitute

for it. Linear Algebra is a fairly abstract theoretical course, and this guide should contain most of the

theory. However, being able to apply the theorems is also important, since it tests your understanding.

Disclaimer:Use at your own risk. No guarantee is made that this revision guide is accurate or complete, or that it will improve your exam performance. Use of this guidewillincrease entropy, contributing to the heat death of the universe. Contains no GM ingredients. Your mileage may vary.

All your base are belong to us.

Authors

Originally by Shriti Somaiya, edited by Dave Taylor. Revised in 2007 by J. A. Elliott (j.a.elliott@warwick.ac.uk), with additions by David McCormick (d.s.mccormick@warwick.ac.uk). Further revisions carried out by Jess Lewton & Guy Barwell (r.g.barwell@warwick.ac.uk) 2012. Any corrections or improvements should be entered into our feedback form at http://tinyurl.com/WMSGuides (alternatively email revision.guides@warwickmaths.org).

Originally based upon lectures given by Derek Holt at the University of Warwick, 2001 and 2006, checked

against subsequent courses.

History

First Edition: 2001.

Second Edition: May 16, 2007.

Current Edition: January 18, 2020.

MA106 Linear Algebra11 Introduction

Linearity pervades mathematics:linear algebrais that branch of mathematics concerned with the study

of vectors, vector spaces, linear maps, and systems of linear equations, and is the language with which

we talk about linearity. It has extensive applications in the natural sciences and the social sciences, since

nonlinear models can often be approximated by linear ones. Linear algebra originated from the theoretical study of the solutions of sets of simultaneous linear equations. Using techniques from linear algebra, problems about systems of linear equations can be reduced to equivalent problems about matrices. For instance

2x+y= 1

x3y= 2is equivalent to2 1 13 x y =1 2

1.1 Number Systems and Fields

In order to talk about such problems in as general a setting as possible, we x a denite starting point

and, assuming nothing else, work from there. Our starting point will be number systems, using the term

as a vague intuitive idea rather than giving any formal denition. The most used number systems in mathematics are: NZQRC natural numbers integers rational numbers real numbers complex numbers A perhaps less well-known example is that of the algebraic numbersAC, i.e. those numbers which are solutions of polynomials with rational coecients:p3;i2A, bute; =2A.

1.2 Axioms for Number Systems

The term \axiom" has a variety of meanings in mathematics. Sometimes it is taken to mean a self-

evident undeniable truth; in other situations it simply refers to anything that is assumed without proof

when developing some theory (e.g. linear algebra). Here it is the latter. Denition 1.1.A number systemKis said to be aeldif it satises the following ten axioms: (A1)+=+, for all;2K(commutativity of addition) (A2) (+) + ), for all;;

2K(associativity of addition)

(A3) There exists 02K, such that+ 0 =, for all2K(existence of zero element) (A4) For each2Kthere exists2Ksuch that+ () = 0 (existence of additive inverses) (M1)=, for all;2K(commutativity of multiplication) (M2) () ), for all;;

2K(associativity of multiplication)

(M3) There exists 12Ksuch that1 =, for all2K(existence of identity element) (M4) For each2Knf0gthere exists12Ksuch that1= 1 (existence of multiplicative inverses) (D) (+) , for all;;

2K(distributivity of multiplication over addition)

(ND) 16= 0 (non-degeneracy1)

Recall the denition of a group:

Denition 1.2.LetGbe a set and letbe a binary operation onG(a map that takes any two elements ofGand returns an element ofG). We say that the pair (G;) is a group if (G0) The setGis closed with respect to the operation, i.e. if;2Gthen2G. (Strictly speaking, this is part of the denition of the binary operation, but is often included anyway.) (G1) () ), for all;;

2G(associativity)

(G2) There exists 12Gsuch that1 = 1=, for all2G(existence of identity) (G3) For each2Gthere exists12Gsuch that1=1= 1 (existence of inverses) It is common practice to refer to \the groupG" with the operation implicit.Gis said to beabelian (orcommutative) if additionally =, for all;2G.1 This condition is simply to exclude the trivial setf0gfrom being a eld.

2MA106 Linear AlgebraUsing the idea of a group we can summarise the denition of a eld as follows.

Denition 1.3.A number systemKis said to be aeldif:

Kis an abelian group under addition;

Kn f0gis an abelian group under multiplication;

Multiplication onKdistributes over addition;

16= 0.

2 Vector Spaces

In applied mathematics vectors are often thought of geometrically, perhaps representing some physical

quantity, such as velocity or momentum; in such cases a vector is considered as something with magnitude

and direction. In pure mathematics, on the other hand, vectors can be treated entirely algebraically and

this is how vectors encountered in linear algebra should be thought of: as mathematical objects that

obey certain rules. One advantage of the algebraic approach is that it is just as easy to study vectors in

ndimensions as it is to study them in two or three. Denition 2.1.Avector spaceover a eldKis a setVtogether with two basic operations, known as vector addition and scalar multiplication, such that the following axioms hold: (V0) The setVis closed under vector addition and scalar multiplication. That is, ifv;w2Vand2K thenv+w2Vandv2V. (As in the denition of a group, this axiom is actually part of the denition of the operations themselves but is included as a reminder.) (V1) With respect to the operation of vector addition,Vis an abelian group. (V2)(v+w) =v+w, for all2K;v;w2V (V3) (+)v=v+v, for all;2K;v2V (V4) ()v=(v), for all;2K;v2V (V5) 1v=v, for allv2V(where 1 is the identity scalar inK) The elements ofKare called scalars and the elements ofVare called vectors. Often, but not always, Greek letters and boldface letters are used for these, respectively. Note that bothKandVhave zero elements, and these are distinct. The zero scalar is 0

K(sometimes just written \0") and the zero vector

is0V(sometimes0). It is usually not important what eldKactually is. Throughout this course it is safe to assume that K=R, but in later courses there are times when it is necessary to haveK=C(e.g. to nd the Jordan Canonical Form of a matrix { seeMA251 Algebra I: Advanced Linear Algebra). Using the axioms of vector spaces it is possible to prove some obvious properties of vectors and scalars, such as

0V=0V, for all2K.

0Kv=0V, for allv2V.

(v) = ()v=(v), for all2K;v2V.

2.1 Subspaces

Another important denition is that of a vector subspace. Denition 2.2.LetVbe a vector space and letWVbe non-empty. We say thatWis a (vectoror linear)subspaceofVifWis itself a vector space with respect to the same operations as those onV.

IfW6=Vthen we say thatWis aproper subspaceofV.

Note that sinceWis a subset ofVmost of the properties ofVare carried over toWso we only

really need to check thatWis closed with respect to the relevant operations. This is summed up in the

following proposition. Proposition 2.3.LetVbe a vector space over a eldKand letWVbe non-empty. If for all v;w2Wand2Kwe havev+w2Wandv2WthenWis a subspace ofV. For any given vector spaceV, the setsVandf0Vgare always automatically subspaces ofV, which we refer to as \trivial subspaces". Note that every subspace ofVmust contain the zero vector0V. MA106 Linear Algebra3Proposition 2.4.IfW1andW2are subspaces of a vector spaceVthenW1\W2andW1+W2= fw1+w2jw12W1;w22W2gare both subspaces ofV. Note thatW1+W2is not the same asW1[W2, which may not even be a subspace. For example, the linesW1=f(;0)j2RgandW2=f(0;)j2Rgare both subspaces ofR2, but their union is not a subspace as it is not closed (e.g. (1;0) + (0;1) = (1;1)=2W1[W2). Denition 2.5.Two subspacesW1andW2of a vector spaceVare said to becomplementaryif W

1\W2=f0VgandW1+W2=V. This is equivalent to saying that each vectorv2Vcan be written

uniquely asv=w1+w2wherew12W1andw22W2.

2.2 Examples of Vector Spaces

The most obvious example of a vector space isKn(sometimes writtenVn(K)), where the vectors are n-tuples of elements ofK. That is, K n=f(1;2;:::;n)j1;2;:::;n2Kg: For instance, ifK=RthenRnis justn-dimensional space (e.g.R2=f(;)j;2Rgis the set of

ordered pairs, representing points in the plane). Vector addition and scalar multiplication are dened in

the obvious way. (1;:::;n) + (1;:::;n) = (1+1;:::;n+n); (1;:::;n) = (1;:::;n):

The zero vector is0= (0;0;:::;0).

Examples of non-trivial subspaces ofRninclude lines, planes, etc. up to (n1)-dimensional hy- perplanes through the origin. Forn= 3, for instance, a line through the origin is a set of the form fvj2Rgfor some direction vectorv. The set of all polynomials with coecients inKand degree less than or equal to some xed natural numbernis a vector space,K[x]n(sometimes writtenPn(K)), where vector addition and scalar multi- plication are dened as expected. In fact the set ofallpolynomials with coecients inK(and unlimited degree) is a vector space,K[x]. However, the set of all polynomials with coecients inKand degree

exactlynis not a vector space as it is not closed under vector addition. For any natural numbern, the

vector spaceK[x]nis a subspace ofK[x]. As an example from analysis, the set of all real-valued functions on some setARis a vector space, with vector addition and scalar multiplication dened by (f+g)(x) =f(x) +g(x) (f)(x) =f(x) The set of continuous real-valued functions dened onA, which we denoteC0(A), is a subspace of this vector space.

3 Linear Independence, Spanning Sets and Bases

An important idea in linear algebra is that of the dimension of a vector space. Geometrically, the dimension can be thought of as the number of dierent \coordinates" (e.g.R3is 3-dimensional as it can

be described by anx-, ay- and az-coordinate). This intuitive interpretation works well for relatively

simple vector spaces, such asRn, but is somewhat less useful for more complicated examples, including

spaces of polynomials or functions. It is possible to dene dimension of a vector space in a purely algebraic way using the notion of a \basis". There are some important preliminary denitions. Denition 3.1.Alinear combinationof a set of vectorsv1;:::;vnin a vector spaceVover a eldK is any sum

1v1++nvn

where1;:::;nare scalars (possibly zero) inK.

4MA106 Linear AlgebraDenition 3.2.A set of vectorsv1;:::;vnin a vector spaceVover a eldKare said to belinearly

independentif none of them is a linear combination of the others. This is the same as saying that

1v1++nvn=0V=)1==n= 0K:

If the vectorsv1;:::;vnare not linearly independent then we say that they arelinearly dependent. Lemma 3.3.A set of vectorsv1;:::;vn2Vare linearly dependent if and only if for somevreither v r=0Vorvris a linear combination ofv1;:::;vr1;vr+1;:::;vn. Denition 3.4.A set of vectorsv1;:::;vnin a vector spaceVover a eldKare said tospanVif

everyv2Vcan be written in at least one way as a linear combination of vectors in the set. That is, if

for allv2Vthere exist scalars1;:::;n2K, such that v=1v1++nvn: Denition 3.5.A set of vectorsv1;:::;vnin a vector spaceVare said to form abasisforVif they are linearly independent and spanV. Proposition 3.6.Ifv1;:::;vnis a basis for the vector spaceVthen everyv2Vcan be written as a uniquelinear combination of the vectorsv1;:::;vn. That is, v=1v1++nvn where the scalars1;:::;nare uniquely determined byv. Theorem 3.7.Any two bases2of a vector space contain the same number of vectors. (append the two bases and apply 3.9)

3.1 Dimension

The previous result means that the following is well-dened. Denition 3.8.Thedimensionof a vector spaceVis the number of vectors in any basis forV. We write dimVfor the dimension ofV. (By convention, dimf0Vg= 0.) For example, dimKn=n; any vector can be described uniquely byncoordinates. Any vector space Vwhere dimV=nfor some natural numbernis said to benite dimensional. There are also vector spaces with innite dimension:K[x] has the countably innite basis

1;x;x2;x3;:::;xn;:::

whereas the space of all real-valued functions dened on a setARhas uncountably innite dimension. However, this course deals almost exclusively with nite dimensional vector spaces. Note that a nite dimensional vector space is not necessarily nite. For instance, consider the plane R

2. This has a nite dimension of two, but contains an uncountably innite number of points (vectors).

As long as the eldKis innite then so is the vector space. Lemma 3.9.Suppose that the vectorsv1;:::;vn;w2VspanVand thatwis a linear combination of v

1;:::;vn. Thenv1;:::;vnspanV. In other words, given a spanning set, you can remove any vector

that is a linear combination of the others and still have a spanning set; this is called \sifting". Corollary 3.10.Suppose that the vectorsv1;:::;vr2VspanVand that dimV=nwherer > n. Then the setfv1;:::;vrgcontains a proper subset that is a basis forV. That is, any spanning set can be reduced to a basis. Lemma 3.11.Suppose thatVis ann-dimensional vector space and that the vectorsv1;:::;vr2Vare linearly independent, wherer < n. Then there exist vectorsvr+1;vr+2;:::;vn2Vsuch thatv1;:::;vn forms a basis forV. Thus, any set of linearly independent vectors can be extended to a basis.2

The plural of basis is \bases".

MA106 Linear Algebra5Given two subspacesW1andW2of a vector spaceV, we can form the subspacesW1+W2and W

1\W2. As any subspace is itself a vector space, we can nd the dimension of each subspace. The

following theorem tells us how the dimensions ofW1,W2,W1+W2andW1\W2are related. Theorem 3.12.Suppose thatVis a nite-dimensional andW1,W2are two subspaces ofV. Then dim(W1+W2) = dim(W1) + dim(W2)dim(W1\W2):

4 Matrices and Linear Maps

4.1 Linear Transformations

Single vector spaces considered in isolation are not very interesting. The main results in linear algebra are

concerned with the maps between vector spaces, which are called linear maps (or linear transformations).

Denition 4.1.LetUandVbe two vector spaces over the same eldK. Alinear map(orlinear transformation) is a mapT:U!Vsuch that

T(u1+u2) =T(u1) +T(u2), for allu1;u22U

T(u) =T(u), for allu2Uand2K.

These two conditions can be condensed into one equivalent condition:

T(u1+u2) =T(u1) +T(u2);for allu1;u22Uand;2K:

Proposition 4.2.The following results follow immediately.

T(0U) =0V.

T(u) =T(u), for allu2U:

Linear maps between vector spaces are just one example of structure-preserving maps between alge- braic structures. A homomorphism between two groups (G;) and (H;) is a map:G!Hsuch that (g1g2) =(g1)(g2) for everyg1;g22G, i.e. such that in some sense the structure is preserved. A linear map between vector spaces can be thought of as a type of homomorphism. There are many examples of linear maps between vector spaces. For instance, the embeddingT:R2! R

3dened byT: (;)7!(;;0) is a linear map, as is a rotation about the origin in the plane (i.e.R2).

However, there are also plenty of examples of maps between vector spaces which are not linear. Consider the translationT:Rn!Rndened byT: (1;2;:::;n)7!(1+ 1;2;:::;n). Since

T(0)6=0, this cannot be linear.

The following theorem is very important.

Theorem 4.3.A linear map is completely determined by its action on a basis. If two linear maps have the same eect on a basis of the domain then they are the same map.

Now, some more denitions.

Denition 4.4.LetUandVbe vector spaces and letT:U!Vbe a linear map. TheimageofT, written imT, is the set of vectorsv2Vsuch thatv=T(u) for someu2U. That is, imT=fT(u)ju2Ug: Denition 4.5.LetUandVbe vector spaces and letT:U!Vbe a linear map. Thekernel(or nullspace) ofT, written kerT, is the set of vectorsu2Usuch thatT(u) =0V. That is, kerT=fu2UjT(u) =0Vg: Proposition 4.6.The kernel and image of a linear mapT:U!Vare subspaces ofUandV, respec- tively. Denition 4.7.Therankof a linear mapT:U!Vis the dimension of its image, i.e. rankT= dim(imT).

6MA106 Linear AlgebraDenition 4.8.Thenullityof a linear mapT:U!Vis the dimension of its kernel, i.e. nullityT=

dim(kerT). The dimensions of the kernel and image of a linear map between vector spaces are closely related.

The next theorem tells us how.

Theorem 4.9(Dimension Theorem).LetUandVbe nite-dimensional vector spaces over a eldK and letT:U!Vbe a linear map. Then rankT+ nullityT= dimU: Proposition 4.10.LetUandVbe vector spaces with dimU= dimV=nand letT:U!Vbe a linear map. Then the following are equivalent.

Tis surjective

rankT=n nullityT= 0

Tis injective

Tis bijective

Denition 4.11.IfUandVare vector spaces with dimU= dimVa linear mapT:U!Vis said to

benon-singularif it is surjective andsingularif it is not. Equivalently, a mapTis singular if kerT6= 0.

Linear maps can be combined in several ways. LetU,VandWbe vector spaces and letT1:U!V, T

2:U!VandT3:V!Wbe linear maps. Then the following are also linear maps:

T1+T2:U!V, dened as (T1+T2)(u) =T1(u) +T2(u), for allu2U. T1:U!V, dened as (T1)(u) =T1(u), for allu2Uand xed2K. T3T2:U!W, dened as (T3T2)(u) =T3(T2(u)), for allu2U.

4.2 Matrices

Matrices are combinatorial structures which represent linear transformations. That is, the eect of multiplying a column vector by a matrix gives the same result as applying the corresponding linear transformation to that column vector, and vice versa. For example, the mapT:R2!R2dened by T:x y 7!x+y 2xy is described by the matrix 1 1 21
, since 1 1 21
x y =x+y 2xy The basic matrix operations are straightforward. Addition and scalar multiplication are carried out

term by term. Slightly more complicated is the multiplication of two matrices. Note that two matrices

can only be multiplied together if the second has the same number of rows as the rst has columns.

Using the notation

(ij) =0 B

111n.........

m1mn1 C A the product ofA= (ij) andB= (ij) isAB=CwhereC= ( ij) and ij=nX k=1 ikkj: MA106 Linear Algebra7Proposition 4.12.The following \laws of matrices" always hold:

A+B=B+AwhereAandBaremnmatrices.

(A+B)C=AC+BCwhereAandBarelmmatrices andCis anmnmatrix. A(B+C) =AB+ACwhereAis anlmmatrix andBandCaremnmatrices. (A)B=(AB) =A(B) whereAis anlmmatrix,Bis anmnmatrix and2K. (AB)C=A(BC) whereAis anlmmatrix,Bis anmnmatrix andCis annpmatrix. Note that matrix multiplication isnotcommutative, i.e. in general it is not true thatAB=BA. Denition 4.13.Thetransposeof anmnmatrixA, writtenAt, is thenmmatrix whose (i;j)th entry is the (j;i)th entry ofA. Proposition 4.14.For anmnmatrixAand anpmatrixB, (AB)t=BtAt. Denition 4.15.Thezero matrixhas every entry equal to 0. Sometimes themnzero matrix is written 0 mn. Denition 4.16.Thennidentity matrixis a squarennmatrix (ij) whereii= 1 for 1in andij= 0 ifi6=j. In other words,ij=ij, the Kronecker delta. LetUandVbe vector spaces and letT:U!Vbe a linear map. For each choice of basis forU andVthere is exactly one matrix that representsT. The product of two matrices representing linear

transformations corresponds to the composition of the linear transformations, provided the multiplication

is carried out in the correct order.

5 Elementary Operations and the Rank of a Matrix

Recall that the rank of a linear mapT:U!Vis the dimension of its image, i.e. rankT= dim(imT).

5.1 Row and Column Operations

Denition 5.1.Therow rankof a matrixAis the dimension of the vector space spanned by the vectors that make up its rows. Denition 5.2.Thecolumn rankof a matrixAis the dimension of the vector space spanned by the vectors that make up its columns. Lemma 5.3.IfT:U!Vis a linear map represented by a matrixAthen rankT= rowrankA= columnrankA:

In order to determine the rank of a matrix it is often easier to row- or column-reduce it. This is done

via a series of operations which do not change the rank and results in a matrix whose rank can be read

o without any thought. LetAbe anmnmatrix. There are threeelementary row operationsand threeelementary column operations. (R1) Add a multiple of one row to another dierent row. (R2) Interchange two dierent rows. (R3) Multiply a row by anon-zeroscalar. The column operations (C1){(C3) are dened analogously to the row operations above. Lemma 5.4.Applying any row operation (R1){(R3) or any column operation (C1){(C3) to a matrix

Adoes not change the rank ofA.

All row and column operations can be represented byelementary matriceswhose eect on another

matrix is then to apply that row or column operation. To obtain the matrix of a row or column operation

simply apply the row or column operation to the identity matrix, giving the elementary matrix required.

The eect of applying the same row or column operation to a matrix is then found by premultiplying

8MA106 Linear Algebrathat matrix by the elementary matrix just found. Make sure you know how to express a matrix in terms

of elementary matrices. By repeated application of the elementary row operations a matrix can be reduced toupper echelon form, where the leftmost non-zero element of each row is equal to 1:0 B

B@1

0 1

0 0 0 1

0 0 0 0 01

C CA: The rank of the original matrix is then equal to the number of non-zero rows in upper echelon form. It is possible to further reduce a matrix in upper echelon form torow-reduced echelon formwhere the

leftmost non-zero entry in each row is a 1 and is the only non-zero entry in that column, for example:0

B@1 2 0 0 0

0 0 1 0 1

0 0 0 1 0

0 0 0 0 01

C CA:

The above matrix has a rank equal to 3.

By applying column operations, we can further reduce a matrix in row-reduced echelon form toSmith

normal form, which has the rstsentries on the leading diagonal equal to 1 and all other entries zero,

for example:0 B

B@1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 0 01

C CA: Proposition 5.5.Letsbe the number of non-zero rows in the Smith normal form of a matrixA. Then both the row rank ofAand the column rank ofAare equal tos.quotesdbs_dbs8.pdfusesText_14

[PDF] MA106 Linear Algebra Revision Guide - Warwick Maths

Linear Algebra

1 Introduction1

2 Vector Spaces2

3 Linear Independence, Spanning Sets and Bases 3

4 Matrices and Linear Maps 5

5 Elementary Operations and the Rank of a Matrix 7

6 Determinants and Inverses 8

7 Change of Basis and Similar Matrices 10

Introduction

All your base are belong to us.

Authors

History

First Edition: 2001.

Second Edition: May 16, 2007.

Current Edition: January 18, 2020.

MA106 Linear Algebra11 Introduction

2x+y= 1

1.1 Number Systems and Fields

1.2 Axioms for Number Systems

2K(associativity of addition)

2K(associativity of multiplication)

2K(distributivity of multiplication over addition)

Recall the denition of a group:

2G(associativity)

2MA106 Linear AlgebraUsing the idea of a group we can summarise the denition of a eld as follows.

Kis an abelian group under addition;

Kn f0gis an abelian group under multiplication;

Multiplication onKdistributes over addition;

16= 0.

2 Vector Spaces

K(sometimes just written \0") and the zero vector

0V=0V, for all2K.

0Kv=0V, for allv2V.

2.1 Subspaces

IfW6=Vthen we say thatWis aproper subspaceofV.

1\W2=f0VgandW1+W2=V. This is equivalent to saying that each vectorv2Vcan be written

2.2 Examples of Vector Spaces

The zero vector is0= (0;0;:::;0).

3 Linear Independence, Spanning Sets and Bases

1v1++nvn

4MA106 Linear AlgebraDenition 3.2.A set of vectorsv1;:::;vnin a vector spaceVover a eldKare said to belinearly

1v1++nvn=0V=)1==n= 0K:

3.1 Dimension

1;x;x2;x3;:::;xn;:::

2. This has a nite dimension of two, but contains an uncountably innite number of points (vectors).

1;:::;vn. Thenv1;:::;vnspanV. In other words, given a spanning set, you can remove any vector

The plural of basis is \bases".

1\W2. As any subspace is itself a vector space, we can nd the dimension of each subspace. The

4 Matrices and Linear Maps

4.1 Linear Transformations

T(u1+u2) =T(u1) +T(u2), for allu1;u22U

T(u) =T(u), for allu2Uand2K.

T(u1+u2) =T(u1) +T(u2);for allu1;u22Uand;2K:

T(0U) =0V.

T(u) =T(u), for allu2U:

3dened byT: (;)7!(;;0) is a linear map, as is a rotation about the origin in the plane (i.e.R2).

T(0)6=0, this cannot be linear.

The following theorem is very important.

Now, some more denitions.

6MA106 Linear AlgebraDenition 4.8.Thenullityof a linear mapT:U!Vis the dimension of its kernel, i.e. nullityT=

The next theorem tells us how.

Tis surjective

Tis injective

Tis bijective

2:U!VandT3:V!Wbe linear maps. Then the following are also linear maps:

4.2 Matrices

Using the notation

111n.........

A+B=B+AwhereAandBaremnmatrices.

5 Elementary Operations and the Rank of a Matrix

5.1 Row and Column Operations

Adoes not change the rank ofA.

8MA106 Linear Algebrathat matrix by the elementary matrix just found. Make sure you know how to express a matrix in terms

B@1

0 0 0 1

0 0 0 0 01

B@1 2 0 0 0

0 0 1 0 1

0 0 0 1 0