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For problem 1-5, determine the null space of
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Differential Equations and Linear Algebra (4th Edition)
- Solve each of the following equations by finding [ a ]1 and using the result in Exercise 9. a.[ 4 ][ x ]=[ 5 ]in13b.[ 8 ][ x ]=[ 7 ]in11c.[ 7 ][ x ]=[ 11 ]in12d.[ 8 ][ x ]=[ 11 ]in15e.[ 9 ][ x ]=[ 14 ]in20f.[ 8 ][ x ]=[ 15 ]in27g.[ 6 ][ x ]=[ 5 ]in319h.[ 9 ][ x ]=[ 8 ]in242 Let [ a ] be an element of n that has a multiplicative inverse [ a ]1 in n. Prove that [ x ]=[ a ]1[ b ] is the unique solution in n to the equation [ a ][ x ]=[ b ].arrow_forwardFind the null space for A. - [1 null(A) A = 15-5 0 1 -³] 3 = spanarrow_forwardFor which value of x is A = x = 4 1 -4 4 0 -3 0 1 x NOT invertible?arrow_forward
- .Let Z2 be an algebraic closure of Z2, and let a, ß e Z2 be zeros of x3 x2+1 and of x3 +x + 1, respectively Using the results of this section, show that Z2(a) = Z2(B)arrow_forward7. The two quadratic relations intersect at 2 points, points A and B. Determine graphically the coordinates of each of the points of intersection. O y (х-1)(х—9) 2 10 ТУ 2- -10-8-6 -4-2 -2+ 4 8 10 -4 -81 -10 1 1 39 4 4 XT do ||arrow_forwardThe sum of two positive real numbers x and y is 3. Find an x and y among these such that the product xy² is maximal. X = y =arrow_forward
- If A is 3 × 3 with rank A = 2, show that the dimension of the null space of A is 1.arrow_forwardLet [3 1 2] A = 2 1 2 [1 2 2] Find the following: 1.1 AT ЗАТ A-1 1.2 1.3 1.4 3AT + A1arrow_forwardSuppose (x1,x2) + (Y1,Y2) is defined to be (x1 + Y2 ,x2 + Y1). With the usual multiplication cx = ( cx1, cx2 ), which of the eight conditions are not satisfied?arrow_forward
- Show that ℝ3 = span( 1 1 2 1 2 1 0 3 -1arrow_forwardDescribe geometrically (line, plane, or all of R 3) all linear combinations of (a) [ 1/2/3 ] and [ 3/6/9 ] (b) [ 1/0/0 ] and [ 0/2/3 ] (c) [ 2/0/0 ] and [ 0/2/2 ] and [ 2/2/3 ]arrow_forwardSuppose that x and y are real numbers such that y is 6 greater than x. What is the smallest possible valueof the product xy?arrow_forward
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