Spring 2009

 Office Hour: Wednesday 5:45--6:45 PM, Thursday 5:45--6:45 PM (510 Fine)

Books

• Gilbert Strang, Linear Algebra and itâ€™s Applications, 4th Ed. (It has a related website: http://web.mit.edu/18.06/www/ )
• Kenneth Hoffman, Ray Kunze, Linear Algebra. (It is a more abstract approach toward linear algebra. This is a standard and well-written book.)

Schedule

You can find a tentative schedule for the course on the "blackboard", or here.

What we have done so far

• First week:
• Geometry of linear equations.
• Length (norm, magnitude) of a vector with n components. Dot product of two vectors.
• Elimination method. Cost of this algorithm.
• Singular and non-singular systems.
• Matrix multiplication.
• Use of blocks in multiplication.
• Second week:
• Elementary matrices.
• LU decomposition (with no changes of rows).
• Use of LU in solving linear equations.
• Permutation matrices.
• PA=LU (with change of rows).
• Sign of a permutation.
• Uniqueness of LDU decomposition.
• Multiplying by a diagonal matrix from left (or right).
• Transpose of a matrix.
• Symmetric matrix.
• (AB)T=BTAT.
• Third week:
• If LDU is symmetric, then L=UT.
• Inverse of a matrix.
• Right inverse=left inverse (if both of them exist).
• Square matrix+ left inverse ----> invertible.
• P-1=PT if P is a permutation martix.
• A-1=AT if A is an orthogonal matrix.
• Eij(c)-1= Eij(-c).
• (AB)-1=B-1A-1 if both A and B are invertible.
• If AB and B are invertible, so is A.
• How to find inverse of a matrix. (only in one of the classes)
• If A is invertible, Ax=0 (column vectors) and yA=0 (row vectors) have unique vector solutions, x=0 and y=0, respectively.
• Necessary and sufficient condition for an upper-triangular matrix to be invertible. (Diagonal entries should be non-zero.)
• A is invertible if and only if AT is invertible. Moreover if they are invertible, (AT)-1=(A-1)T.
• PA=LU; A is invertible if and ony if all the diagonal entries of U are non-zero.
• If A is not invertible, then there is a non-zero vector x such that Ax=0.
• Definition of a vector space.
• Examples of vector spaces from "linear" differential equations, e.g.
{f:[0,1] &rarr R | f is differentiable, f''(x)-f'(x)-f(x)=0}.
• Examples of vector spaces from "linear" recursive sequences, e.g.
 {{an} ∞ n=1 |an's real numbers & an+2=an+1+an for any positive integer n}.
• Linear transformation TA(x)=Ax associated with a given matrix A.
• Image of TA=the column space of A=linear combination of columns of A.
• Null space (or kernel) of A.
• Echelon form of A.
• Using elemination process to describe the column space and the null space of A.
• How to describe "preimage of a vector b under TA", i.e. solutions of Ax=b.
• Forth Week:
• Reduced row echelon form of a matrix.
• How to describe solutions of Ax=b, using rref(A).
• Ax=b has either 0, 1, or infinitely many solutions (revisited).
• If xp is a solution of Ax=b, then any soltuion of this system of linear equations is of the form xp+xn, where xn is a vector in Null(A) the null space of A.
• If A is n by m and m>n, then Null(A) is non-zero, i.e. if the number of variables is more than the number of equations in Ax=0, then there is a non-zero solution.
• Let A be square matrix; then A is invertible if and only if Null(A)={0}.
• Rank(A)= number of leading 1's in the rref(A).
• Linearly independent vectors.
• v1, v2, ... , vn are linearly independent if and only if the null space of A=[v1 ... vn] is zero.
• Columns of an invertible matrix are linearly independent.
• Rows of an invertible matrix are linearly independent.
• Spanning sets.
• Columns of a matrix span the column space.
• Define a basis.
• dim (Rn)=n.
• How to find a basis of the null space and the column space.
• Null(A)=Null(rref(A)).
• dim (column space of A)=rank(A).
• Fifth week:
• row space of A=row space of rref(A).
• dim (row space of A)= rank(A).
• rank(A)=rank(AT).
• dim (left null space of A)= #rows-rank(A).
• dim (Null space of A)= #columns-rank(A).
• Let T:Rn &rarr Rm be a linear transformation; Then
dim(Rn)=dim(ker(T))+dim(Im(T)).
• If A has a right-inverse, then the left-null space of A is zero.
• If A has a left-inverse, then the null space of A is zero.
• Left-null space of A is zero if and only if rank(A)= #rows.
• Null space of A is zero if and only if rank(A)= #columns.
• If rank(A)= #rows, then A has a right-inverse.
• The following statements are equivalent:
• A has a left inverse.
• N(A)={0}.
• rank(A)=# columns.
• If Ax=b has a solution, it is unique.
• The row space = R# columns.
• For any b, xTA=bT has a solution.
• AT has a right inverse.
• Any finite dimensional (real) vector space V can identified with Rdim V.
• Let B={v1,...,vn} be a basis of V. For any v in V, let
[v]B=[c1 ... cn]T,
where v=c1 v1+...+cnvn. Then [.]B:V &rarr Rn is a 1-1 and surjective linear map.
• Let B1={v1,...,vn} and B2={w1,...,wn} be two basis of V. Then the following diagram is commutative:
where S=[[w1]B1...[wn]B1], i.e. [v]B1=S [v]B2.
• Let T be a linear transformation from V to W, B={v1,...,vn} a basis of V, and B'={w1,...,wm} a basis of W. Then the following diagram is commutative:
where A=[[T(v1)]B'...[T(vn)]B'], i.e. [T(v)]B'=A [v]B.
• Sixth week:
• The following diagram is commutative:
where S, S', A1, and A2 are similar as above. It is the same as saying that
A2=S'-1 A1 S.
• Let T be a linear transformation from V to V. Let B1 and B2 be two basis of V. In the above setting, let W=V, B'1=B1, and B'2=B2; then
S'=S and A2=S-1 A1 S.
• We say that A is similar to B if A=S-1 B S, for some S.
• Description of matrix of rotation in the standard basis.
• Description of matrix of projection onto a line and more generally onto a direction in Rn in the standard basis.
Proja(v)=(a.v/a.a)a=(1/aTa) a (aTv)=(1/aTa) (a aT) v.
So in the standard basis, its matrix is (1/aTa) (a aT), which is a symmetric matrix.
• Remark: since we used the dot product for the projection, we are writing v in the standard basis (or orthonormal basis which we will learn later).
• Description of matrix of reflection about a line which passes through the origin.
RefL(v)+v=2 ProjL(v),
therefore
RefL(v)=2 ProjL(v)-v,
and we get what we wanted.
• Changing the standard basis and showing that the projection matrix is similar to
 1 0 0 0
and the reflection matrix is similar to
 1 0 0 -1
• Othogonal complement of a subspace.
• Let V be a subspace of Rn; (V)=V.
• Let V be a subspace of Rn; then Rn=V ⊕ V, i.e. any vector b in Rn can be written as a sum of a vector v in V and v in V, in a unique way.
• If b=v+v, where v is in V and v is in V, then ProjV(b)=v and ProjV(b)=v.
• ProjV+ProjV=idRn.
• Let V be a subspace of Rn, x and y two vectors in Rn; then x=ProjV(y) if and only if
• x is in V,
• y-x is in V.
• C(A)=N(AT). (Similarly, C(AT)=N(A).)
• Let A be an n by m real matrix, TA the associated linear transformation from Rm to Rn; then restriction of TA is an isomorphism between the row space C(AT) and the column space C(A), i.e. it is a linear map whose kernel is zero and whose range (image) is the co-domain.
• Getting the best possible answer to an inconsistent system of linear equations Ax=b.
• Normal equation associated with a least-square problem: ATA x=ATb.
• N(ATA)=N(A).
• C(ATA)=C(AT).
• If columns of A are linearly independent, then
• ||Ax-b|| is minimize at
x= (ATA)-1AT b.
• Matrix P of ProjC(A) the orthogonal projection onto the column space of A in the standard basis is equal to
A(ATA)-1AT
• Seventh week:
• Linear least square method.
• The same method works for the other kind of expected functions, i.e. as long as we expect that our output is a linear combination of a finite set of given functions. For example, if we expect to get a polynomial of degree n, then A=[aij] such that aij=tij-1.
• If P2=P and PT=P, then it represents an orthogonal projection onto C(P).
• If P is projection onto a subspace V, then C(P)=V.
• We say that a map T:Rn→ Rm preserves the Euclidean structure if
• ||T(x)-T(y)||=||x-y|| for any x and y.
• Angle between the segments T(x)T(y) and T(y)T(z)= angle between the segments xy and yz.
• If T:Rn→ Rm is a linear map which preserves the Euclidean structure and Q is the matrix associated with T in the standard basis, then
• Columns of Q have length 1.
• Columns of Q are pairwise orhtogonal to each other.
• If Q is an m by n matrix such that
• Columns of Q have length 1 and
• Columns of Q are pairwise orhtogonal to each other,
then
• QTQ=In,
• (Qx).(Qy)=x.y,
• ||Qx||=||x||,
• ∠(Qx,Qy)=∠(x,y).
• Let T:Rn→Rm be a linear transformation. The following statements are equivalent:
• ||T(x)||=||x|| for any x.
• T(x).T(y)=x.y for any x and y.
• ||T(x)||=||x|| and ∠(T(x),T(y))=∠(x,y).
• Definition of orthonormal, orthonomal basis and orthogonal matrices.
• QTQ=In if and only if the columns of Q are orthonormal.
• If Q is a square matrix and its columns are orhtonormal, then its rows are also orthonormal.
• If qi's are non-zero and pairwise orhtogonal to each other, and
b=c1q1+...+cnqn,
then
ci=(qi.b)/(qi.qi).
• If ai's are linear independent, how we can find the projection of a given vector onto the span of ai's. (Let A be a matrix whose columns are ai's and then use A(ATA)-1AT.)
• We can also find an orthonormal basis for the of span of ai's and then use dot product.
• If the columns of Q are orthonormal, then the matrix of projection onto the C(Q) in the standard basis is QQT.
• Gram-Schmidt process, i.e. let ai be linearly independent vectors.
•  bj:=aj-&Sigma j-1i=1 (qi.aj)qi
• qj=bj/||bj||,
then qi's are an orthonomal basis of the span of ai's.
• How to use Gram-Schmidt to find an orthonormal basis of span of given vectors.
• If the columns of A are linearly independent, then A=QR, where Q has orthonormal columns and R is an upper-triangular matrix. (A is not necessarily a square matrix.)
• If columns of A are ai's, and qi's are the outcomes of the Gram-Schmidt process, then the for i at most j, the ij entry of R is qi.aj.
• Eighth week:
• What a ring is.
• det:Mn(R) → R s.t.
• det(In)=1.
• Exchange of rows changes the sign.
• Linear w.r.t. the first row.
• Linear w.r.t. any row.
• Equal rows ⇒ det=0.
• (resticted) Row operations does not change.
• Zero row ⇒ det=0.
• A triangular ⇒ det(A)=product of its diagonal entries.
• A is singular iff det(A)=0.
• PA=LU ⇒ det(A)=sgn(P) product of diagonal entries of U.
• Product rule: det(AB)=det(A)det(B).
• det(A)=det(B) if A and B are similar.
• det(T) is well-defined when T:V→ V is a linear transformation.
• Transpose rule: det(A)=det(AT).
• "Big formula"  det(A)=Σ s in Sn sgn(s) Π n i=1 ais(i).
•  det || A0 BC || = det(A) det(C).
• Aij: (n-1) by (n-1) matrix after deleting the i th row and the j th column of A.
• For any i between 1 and n,  det(A)=Σ n j=1 (-1)(i+j) det(Aij) aij
• For any i and k between 1 and n, if k is not equal to i,  0=Σ n j=1 (-1)(i+j) det(Aij) akj
• The ij entry of the adjoint of A is (-1)(j+i) det(Aji).
• Using transpose: adj(A) A=det(A) In.
• If A is invertible, A-1=adj(A)/det(A).

Homework

You can find a list of your weekly problem sets here . Problems are due at 2:00 pm, on Fridays. You can put them in the correct envelope in the box in front of my office. Do not forget to staple them and write down the time of your class.

Extra problems (optional)

• (Cauchy-Schwartz inequality) Let v and w be two vectors with n real components. Show that:
|v.w| ≤ ||v|| ||w||.
Hint: Consider 0 ≤ ||v+cw||2 for any scalar c.
• Why is sign of a permutation well-defined?
• Can the following Sixteen puzzle be solved?
 2 1 3 4 5 6 7 8 9 10 11 12 13 14 15
• Show that:
• a) If L1 and L2 are two lower triangular matrices, so is L1L2.
• b) If P1 and P2 are two permutation matrices, so is P1P2.
• c) If P is a permutation matrix, then so is PT. Moreover PPT=PTP=In.
• Show that if A is a square matrix with a right inverse, then it is invertible.
• Show that if an upper-triangular matrix has non-zero entries on the diagonal, then it is invertible. Moreover its inverse is also upper-triangular. (The same is true for a lower-triangular matrix)
• Let V be the following vector space,
 {{an} ∞ n=1 |an's real numbers & an+2=an+1+an for any positive integer n}.
• Show that any sequence in V can be determined uniquely by its first two terms a1 and a2.
• Use the first part to show that if {an} and {bn} are two non-zero elements of V such that one of them is not a multiple of the other one, then any element of V can be written as a linear combination of {an} and {bn}.
• Show that there are (only) two real numbers α and β such that an=&alphan and bnn are in V.
• Show that for any {an} in V, one can find real numbers A and B such that
an=A &alphan+ B &betan,
for any positive integer n.
• Show that if Null(A)={0}, then # columns of A= rank(A).
• Show that there are
# columns-rank(A)
many vectors such that any vector in Null(A) is a linear combination of them.