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M2solutions225 - Question 1 If the reduced echelon form of...

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Unformatted text preview: Question 1 If the reduced echelon form of 1—23—425 —205—40 A: —2 4 *1—76—9 is 0 0®—3 2 0 1—24—7411 00000® —36—6305 000000 gweg P: u o F: T he rank of A EM: #pxC/ok- ‘* 3 (b) The dimensions of the subspaces Nul(A), Col(A) and R0w(A) CL)“ NAM/H 5 #columm- {Lo/J» 2 (9’3 2 ’3 (we MM} 2 W = 3 M MM) : W: 3 a (c) Bases for R0w(A) and Col(A). 1 0 0 -1 o (g 1 O ) -3 ) 0 .fi 7. 0 o 0 ‘ n» E ,[t . 3 5‘ -z. “I "f \ ) LI , u 3 ‘7 8’ 1 (3) i i x 1 § 1 i' E 2 I i Question 2 Let A be an m X n matrix and suppose that the matrix equation A11: 2 b has solutions for all b E R7”. (a) Find the dimension of COM/1). (2) era/H: QM =2) MCMCM=M (b)Find the dimension of Col(AT) (2) camp) (Lath):7AA/'vvx(€r€c%l7ll z M£Méml M LWD= W“ ((3) Hence compute the dimension of N ull(A ) (2) OW mum) MW (MW): ,7 gum NwUPrT) '5 0 (d) What can you conclude about the solutions to the homogeneous equation ATm = 0? Question 3 Complete the definitions: (a) A set of vectors 5 = {771), U; ..... 27;} in V generates V if (5) vngQ/h xm)-‘ Vial (1". avg rf’uflr, UK]. (a) The set S is a basis for V if «S WV>W «gemWH‘k “iww WGVWL’LMVWMRWM'W Question 4 1 O LetA= 0 1 4 —3 £03100 ottolo 0‘3’9'Ll0' (03100 ,__.7 otLOlO oOl-ng‘ Question 5 Use Cramer’s rule to find 262 if A = < [Useful fact: |Al = 20 ] 1 —1 4 2 —2 3 and A 3 1 5 Question 6 Indicate True (T) 07" false (F) [no reasons need be given]. 7: (a) A line in R4 is a one—dimensional subspace of R4 (b) If vector space V is 23—dimensional and S is a set of 23 non—zero vectors in V, then S is a basis for V. F (c) If the null space of a 7 X 6 matrix A has dimension 4 then dimCol(A) = 2. (d) The columns of a matrix A form a basis for the column space col(A). TX (e) If vector space V contains a set of 9 linearly independent vectors then the dimension of V is at least 9. ”T (f) Switching two rows in an n X n matrix has no affect of the determinant of the matrix. F (g) Elementary row operations don’t change the column space of a matrix. F (b) If A is an n x n matrix and the equation Af : 5 has no non—trivial solutions, then A is invertible. T (i) If A and B are invertible 5 X 5 matrices then so is AB. T (j) If A and B are both 9 X 9 matrices then det(A + B) = det(A) + det(B). 6 i: Question 7 For each of the following, draw a sketch of H and explain either why it is or is not a subspace of R2. (a)H={($,y)€Rzly20} (3) (J) Question 8 Let H be the subspace of R4 defined by H={(:c1,a:2,m3,a:4) I x1—2x2+7a:3+$4=0and m1+3x3=0 }. (a) Find a 2 X 3 matrix A such that H = Nul(A) (4) 319A:\~7,7r\ {030 X \“7/X L'Tqfiyt'Xq C O K e m A A: be :0qu WWW (b) Find a basis for H. XL: ZXSF": X\ '" “3X3 I W O ‘3 t/L X: X3 %1+%H O I O 10 ...
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