Review: matrix multiplication• Ax is a linear combination of the columns of A with weights given by the entriesofx.2 33 1·21= 223+ 131=77• Ax = b is the matrix form of the linear system with augmented matrix [A b].2 33 1·x1x2=b1b22x1+3x2= b13x1+x2= b2• Each column of AB is a linear combination of the columns of A with weights givenby th e corresponding column ofB.2 33 1·2 11 0=7 27 3• Matrix multiplication is not commu t ative: usually, ABBA.A comment on lecture notesMy personal suggestion:• before lecture: have a quick look (15min or so) at the pre-lecture notes to see wherethings are going• during lecture: take a minimal am ount of notes (everyth ing on the screens will bein th e post-lecture notes) and f ocus on t he ideas• after lecture: go through the pre-lecture notes again and fill in all the blanks byyourself• then c ompare with the post-lecture notesSince I am writing the pre-lecture notes a week ahead of time, there is usua lly some minordiffe rences to the post-lecture notes.Armin [email protected] of a matrixDefinition 1. The transpose ATof a matrix A is the matrix whose co lumns are formedfrom the corresponding rows ofA. rows ↔ columnsExample 2.(a)2 03 1−1 4T=2 3 −10 1 4(b) [x1x2x3]T=x1x2x3(c)2 33 1T=2 33 1A matr ix A is called symmetri c if A = AT.Theorem 3. Let A, B be matrices of appropriate size. Then:• (AT)T= A• (A + B)T= AT+ BT• (AB)T= BTAT(illustrated by last practice problem s)Example 4. Dedu ce that (ABC)T= CTBTAT.Solution. (ABC)T= ((AB) C)T= CT(AB)T= CTBTATArmin [email protected] to matrix multipli catio nReview. Each column of AB is a linear combination of the columns of A with weightsgiven by the corresponding column ofB.Two more ways to look at matrix multiplicationExample 5. What is the entry (AB)i,jat row i and column j?Thej-th column of AB is the vector A · (col j of B).Entryi of that is (row i of A) · (col j of B). In other words:(AB)i,j= (row i of A) · (col j of B)Use t his row-column rule to compute:2 3 6−1 0 1·2 −30 12 0=16 −30 32 −30 12 02 3 6−1 0 116 −30 3[Can you see the rul e (AB)T= BTATfrom here?]Observe the symmetry between rows and column s in th is rule!It follows that the interpretation“Each c olumn ofA B is a linear co mbination of the columns of A withweights given by the corresponding column of B.”has the counterp art“Each row ofAB is a linear combination of the rows of B with weights givenby th e corresponding row of A.”Example 6.(a)−1 0 00 0 1·1 2 34 5 67 8 9=−1 −2 −37 8 9Armin [email protected] decompositionElementary matricesExample 7.(a)1 00 1a bc d=a bc d(b)0 11 0a bc d=c da b(c)1 0 00 2 00 0 1a b cd e fg h i=a b c2d 2e 2fg h i(d)1 0 00 1 03 0 1a b cd e fg h i=a b cd e f3a + g 3b + h 3c + iDefinition 8. An elementary matrix is one that is obtaine d by performing a singleelementary row operation on a n identity matrix.The result of an elementary row operation on A is EAwhe re E is an elementary matrix (namely, the one obtained by performing the same row operationon the appropr iate identity matrix).Example 9.(a)1 0 00 1 0−3 0 11 0 00 1 03 0 1=111We write1 0 00 1 03 0 1−1=1 0 00 1 0−3 0 1, but more on inverses soon.Elementary matrices are in vertible because elementary row operations are reversible.(b)1 0 02 1 00 0 1−1=1 0 0−2 1 00 0 1(c)1 0 00 2 00 0 1−1=1121(d)1 0 00 0 10 1 0−1=1 0 00 0 10 1 0Armin [email protected] ce problemsExample 10. Choose either column or row interpretation to “see” the result of thefollowing products.(a)1 2 02 1 20 2 1·1 0 00 1 00 −1 1=(b)1 0 00 1 00 −1 1·1 2 02 1 20 2 1=Armin
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