SOME LINEAR ALGEBRA FACTSHere are some linear algebra facts that you should know for Math 5620. The mostimportant ones will be reviewed in more detail in class (for example the differentdefinitions of A invertible). Please also look at the class text book §6.3 and §6.4.• Let A be a n × m matrix. It has n rows and m columns and its elementsare often written:A =a11a12· · · a1ma21a22· · · a2m............an1an2· · · anm= [aij]i=1...n,j=1...m.• To say A is a real n × m matrix we also use the notation A ∈ Rn×m.• A vector is a n × 1 matrix. For example x ∈ Rnwritten entrywisex =x1x2...xn.• The transpose of a n × m matrix A is a m × n matrix defined as follows:(AT)i,j= (A)j,i, for i = 1, . . . , n and j = 1, . . . , m.• We use the notation ej∈ Rnfor the j−th canonical basis vector of Rn,that isej= [0, . . . , 0, 1|{z}j−th position, 0, . . . , 0]T• The canonical basis vectors are useful for example for picking up the i, j−thentry of a matrix:ai,j= eTiAeTj.• Let A ∈ Rn×m, x ∈ Rm. The matrix vector product y = Ax ∈ Rncan bewritten componentwise:(Ax)i=mXj=1aijxj.The matrix vector product can also be written in terms of the columns. Sodenote by ai∈ Rnthe i−th column of A, that is we partition A as:A =| | · · · |a1a2. . . am| | · · · |ThenAx =mXj=1xiai12 SOME LINEAR ALGEBRA FACTS• For a n × n real matrix A we haveA invertible⇔ det A 6= 0⇔ A non-singular⇔ ker A = {0}⇔ Ax = 0 ⇒ x = 0⇔ The columns of A are linearly independent⇔ There exists a unique inverse A−1• If A ∈ Rn×pand B ∈ Rp×mthen the matrix-matrix product AB ∈ Rn×mis defined as follows(AB)i,j=pXk=1aikbkj, for i = 1, . . . , n and j = 1, . . . , m.• In general the matrix matrix product is not commutative. Before evenconsidering commuting the dimensions have to agree so A and B must besquare. And for square matrices in general AB 6= BA.• The dot (or inner) product between Rnvectors x and y is the scalarxTy =nXi=1xiyi.• The entry (AB)ijof the matrix product AB is the dot product betweenthe i−th row of A and the j−th column of