For which values of t is each set linearly independent? a S={(t,0,0),(0,1,0),(0,0,1)} b S={(t,t,t),(t,1,0),(t,0,1)}

Let f1(x)=3x and f2(x)=|x|. Graph both functions on the interval 2x2. Show that these functions are linearly dependent in the vector space C[0,1], but linearly independent in C[1,1].

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 ].

Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).

Let x and y be in Z, not both zero, then x2+y2Z+.

(5) Which of the following sets are linearly independent. Show all work. set 1:[1, e, ex] or set 2:[e, e, cosh x]

Suppose S = {r, u, d} is a set of linearly independent vectors. If x = 3r+ 5u + 2d, determine whether T = {r, u, x} is a linearly independent set. Select an Answer 1. Is I linearly independent or dependent? If I' is dependent, enter a non-trivial linear relation below. Otherwise, enter 0's for the coefficients. r+ u+ x = 0.