Prove that span({x}) = {ax: a F} for any vector x in a vector space. Interpret this

Label the following statements as true or false. (a) The zero vector is a linear combination of any nonempty set of vectors. (b) The span of 0 is 0 . (c) If 5 is a subset of a vector space V, then span(5) equals the intersection of all subspaces of V that contain S. (d) In solving a system of linear equations, it is permissible to multiply an equation by any constant. (e) In solving a system of linear equations, it is permissible to add any multiple of one equation to another. (f) Every system of linear equations has a solution.

Solve the following systems of linear equations by the method introduced in this section. (a) (b) (c) (d) (e) (f) 2si - 1x2 - 3x3 = - 2 3xi 3x2 2x3 + 5x4 = 7 X\ X2 2x3 x4 = 3 3xi - lx2 + 4x3 = 10 x\ - 2x2 + X3 = 3 2x\ X2 2x3 = 6 x\ + 2x2 X3 + a*4 = 5 xi 4- 4x2 - 3x3 3x4 6 2xi 4- 3x2 X3 4- 4x4 = 8 xi 4- 2x2 + 2x3 = 2 xi + 8x3 + 5.T4 = - 6 xi 4- x2 4- 5.T3 + 5x4 = 3 Xi + 2x2 - 4x3 - X4 + x5 = 7 -Xi 4- l()x3 - 3x4 - 4x5 = -16 2xi + 5x2 - 5x3 ~ 4x4 - x5 = 2 4xi + llx 2 - 7x3 - IOX4 - 2x5 = 7 Xi 4- 2x2 + 6x3 = - 1 2xi + x2 + x3 = 8 3xi + x2 - X3 = 15 xi 4- 3x2 + IOX3 = - 5

For each of the following lists of vectors in R3 , determine whether the first vector can be expressed as a linear combination of the other two. (a) (-2,0,3),(1,3,0),(2,4,-1) (b) (1,2, -3), (-3,2,1), (2,-1,-1 ) (c) (3,4,1), (1,-2,1), (-2,-1,1 ) (d) (2,-1,0), (1,2,-3), (1,-3,2) (e) (5,1,-5), (1,-2,-3), (-2,3,-4 ) (f) (-2,2,2), (1,2,-1), (-3,-3,3 )

For each list of polynomials in P3(K), determine whether the first polynomial can be expressed as a linear combination of the other two. (a) (b) (c) (d) (e) (f) x 3 - 3x 4- 5, x3 4- 2x2 - x + 1, x3 + 3x2 - 1 4x3 + 2x2 - 6, x3 - 2x2 + 4x + 1,3x3 - 6x2 + x + 4 -2x3 - 1 lx2 4- 3x 4- 2, x3 - 2x2 + 3x - 1,2x3 + x2 + 3x - 2 x 3 4- x2 + 2x 4-13,2x3 - 3x2 + 4x + 1, x3 - x2 + 2x 4- 3 x 3 - 8x2 + 4x, x3 - 2x2 + 3x - 1, x3 - 2x + 3 6x3 - 3x2 4- x + 2, x3 - x2 + 2x 4- 3,2x3 - 3x 4-

In each part, determine whether the given vector is in the span of S. (a) (2,-1,1), S = {(1,0,2), (-1,1,1)} (b) (-1,2,1), S = {(1,0,2), (-1,1,1)} (c) (-1,1,1,2), S = {(1,0,1,-1), (0,1,1,1)} (d) (2,-1,1,-3), S = {(1,0,1,-1), (0,1,1,1)} (e) -x 3 4-2x2 4-3x 4-3, S = {x3 4- x 2 4-x + l,x2 4- x 4- 1,X 4- 1} x + 3, S= {x3 + x 2 + x + l,x2 (f) 2x 4-x 4- l,x4- 1} 3 - x 2 (g) 1 2 - 3 4 (h) 1 0 0 1 s = S = 1 0 - 1 0 1 0 1 0 0 1 0 1 1 1 0 0 0 1 0 1 1 1 0 0

Show that the vectors (1,1,0), (1,0,1), and (0,1,1) generate F3 .

In Fn , let ej denote the vector whose jt h coordinate is 1 and whose other coordinates are 0. Prove that {ei, e 2 ,... , en} generates F

Show that Pn(F) is generated by {1, x,... , x n }

Show that the matrices 1 0 0 0 generate M2X2(F)-

Show that if 1 0 0 1 0 0 0 0 1 0 and 0 0 0 1 Mi = 0 0 M2 = 0 0 0 1 and M3 = 0 1 1 0 then the span of {Mi, M2, M3} is the set of all symmetric 2x 2 matrices.

Prove that span({x}) = {ax: a F} for any vector x in a vector space. Interpret this result geometrically in R3 .

Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W

Show that if Si and S2 are subsets of a vector space V such that Si C 52 , then span(Si) C span(52). In particular, if Si C 52 and span(Si) = V, deduce that span(52) V.

Show that if Si and S2 are arbitrary subsets of a vector space V, then span(S'iU5, 2) = span(5i)+span(S'2). (The sum of two subsets is defined in the exercises of Section 1.3.)

Let S\ and S2 be subsets of a vector space V. Prove that span (Si DS2) C span(Si) n span(S2). Give an example in which span(Si D S2) and span(Si) nspan(S2 ) are equal and one in which they are unequal.

Let V be a vector space and S a subset of V with the property that whenever v\, 1)2,... ,vn S and aiVi 4- a2v2 4- 4- anvn = 0, then o\ = 0,2 = = an = 0. Prove that every vector in the span of S can be uniquely written as a linear combination of vectors of S.

Let W be a subspace of a vector space V. Under what conditions are there only a finite number of distinct subsets S of W such that S generates W?