Question

linear algebra

Let V (71, 72, 3}, where 71 73=(2,0,3). (1,3,-1), 2 = (0, 1,4), and (a) Prove: V is a basis. (b) Find the coordinates of (b, b2, bs) with respect to V = {71, U2, 3,}. (c) Suppose M and M' are matrices whose columns span the same vector space V. Let b be the coordinates of relative to M. Write a matrix equation that gives b', the coordinates of relative to M'. (Your answer should be a formula in the form Let U be a matrix where u, =0 if i > j, and L be a matrix where j=0 if i < j. (a) U is called an upper triangular matrix and L is a lower tri- angular matrix. Explain why (b) Prove or disprove: The sum of two upper triangular matrices is upper triangular matrix. (c) Prove or disprove: The product of two upper triangular matrices an upper triangular matrix. is an Let A be a 3 x 3 matrix with entries a, and consider the double wide matrix a12 a13 a11 a12 a13 a21 a22 a23 a21 a22 a31 a32 a33 a31 a32 a33 (a) We can view the determinant of the 2 x 2 matriy (b b12 the difference between the "down right" diagonal product bub22 and the "down left" diagonal product byib21 Prove or The determinant of a 3 x 3 matrix can as b21 b22 disprove: also be found by find- ing the difference between the "down right" diagonal products anaz2a33, a12a23a31, a13a21a32, and the "down left" diagonal prod- ucts aina23a32, a12a21a33, a13a22 a31 Suggestion: Find the deter- minant of the 3 x 3 matrix using the cofactors. Then prove (or disprove) that you get the same result using the diagonal products.

Answer 1

V 3,-L - o,+ V3 <20,3> 2 3 65s t s カ 1 J 2 2. 6 L Neml form 29 Coleums ar The atoou mabin has full rank marly indfandut mdshendeut s a bass fir amy 3-dimanslond SV V V are lmady Veclor re Ve y, 2> X Ve f Yt 2 U 2 2X[L3-1] y[o 220,3 2+22, 3 +9. -1 t4y 32 A H いの

31ty 4y32 b3 4y fs2 363 +62 6 b3 +562 961 29 but63 196-95+ 599 2 9 22 Va be a ha 3. V,VVi Xla Un l analhi bo ain Un VL qun n Vn M6alal Chorge

Mv Changs e fe la u T)= Alao from B B2-uv -1 Tlu MN M eotranlaha wat B2 Some LE Xchas a eresmdalimi wot Bg Hon M han MN Mu Would le ike mabi Asn a23 a71 a34 an-Ln yor liangulan faul t yat all lemak are Ubus ile nem

と」た。 ! D a 32 EL Gun ACl laelems l lowr iaguh tr Chriw bthe name hen barguer mabica is oluuy of s Jhe Sn Sum cfleue Lainryuiber mal ot A lu an uthn bn s lu aneli fher meb (a.rthiso =0 A+8 cffr A mabn

6- ais kbis (i, elemit Lij= 0 i>j (As)iT tal air bri k-1 AB (<2i kal ai Ka i ABid ufer A mabi iD 2