Recall from Example 3 in Section 1.3 that the set of diagonal matrices in M2x2(^) is a

Label the following statements as true or false. (a) If S is a linearly dependent set, then each vector in S is a linear combination of other vectors in S. (b) Any set containing the zero vector is linearly dependent. (c) The empty set is linearly dependent. (d) Subsets of linearly dependent sets are linearly dependent. (e) Subsets of linearly independent sets are linearly independent. (f) If aixi 4- a2x2 4- 4- anxn = 0 and Xi,x2 ,... ,xn are linearly independent, then all the scalars a are zero.

Determine whether the following sets are linearly dependent or linearly independent. 1 - 3 - 2 4 (a) - 2 6 4 - 8 in M2x2 (fl) ^ H- l A) \ 2 _ 4 J)- M 2 > < 2 ^ (c) {x3 4- 2x2 , -x 2 4- 3.x 4- 1, x 3 - x 2 4- 2x - 1} in P3(R) 3The computations in Exercise 2(g), (h), (i), and (j) are tedious unless technology is used. Sec. 1.5 Linear Dependence and Linear Independence 41 (d) {x3 - x, 2x2 4- 4, -2x 3 4- 3x2 4- 2x 4- 6} in P3(R) (e) {(l,-l,2),(l,-2,l),(l,l,4)}inR3 (f) {(l,-l,2),(2,0,l),(-l,2,-l)}inR3 (g) (h) 1 0 ~2 1 1 0 ~2 1 0 - 1 1 1 0 - 1 1 1 - 1 2 1 0 - 1 2 1 0 2 1 - 4 4 2 1 2 - 2 in M2x2(tf) in M2x2 (#) (i) {x4 - x 3 4- 5x2 - 8x 4- 6, -x 4 4-x3 - 5x2 + 5x - 3, x 4 4- 3x2 - 3x 4- 5,2x4 4- 3x3 4- 4x2 - x 4-1, x 3 - x 4- 2} in P4(R) (j) {x4 - x 3 + ox2 - 8x 4- 6, -x 4 4- x 3 - 5x2 4- 5x - 3, x 4 + 3x2 - 3x 4- 5,2x4 4- x 3 4- 4x2 4- 8x} in P4(R)

In M3x2 (F), prove that the set is linearly dependent

In Fn , let ej denote the vector whose jth coordinate is 1 and whose other coordinates are 0. Prove that {ei,e 2 ,... ,e n } is linearly independent.

Show that the set {l,x,x 2 ,.. . ,x n } is linearly independent in Pn (F).

In MmXn (F), let E1 * denote the matrix whose only nonzero entry is 1 in the zth row and jth column. Prove that \Ex i: 1 < i < m, 1 < j < n) is linearly independent.

Recall from Example 3 in Section 1.3 that the set of diagonal matrices in M2x2(^) is a subspace. Find a linearly independent set that generates this subspace.

Let S = {(1,1,0), (1,0,1), (0,1,1)} be a subset of the vector space F3 . (a) Prove that if F = R, then S is linearly independent. (b) Prove that if F has characteristic 2, then S is linearly dependent

Let u and v be distinct vectors in a vector space V. Show that {u, v} is linearly dependent if and only if u or v is a multiple of the other

Give an example of three linearly dependent vectors in R3 such that none of the three is a multiple of another.

Let S = {ui,U2, ,Wn} be a linearly independent subset of a vector space V over the field Z2. How many vectors are there in span(S)? Justify your answer.

Prove Theorem 1.6 and its corollary

Let V be a vector space over a field of characteristic not equal to two. (a) Let u and v be distinct vectors in V. Prove that {u, v} is linearly independent if and only if {u 4- v, u v} is linearly independent. (b) Let u, v, and w be distinct vectors in V. Prove that {u,v,w} is linearly independent if and only if {u + v,u + w, v 4- w} is linearly independent.

Prove that a set S is linearly dependent if and only if S = {0} or there exist distinct vectors v,U\,U2, , un in S such that v is a linear combination of ui,u2,...,un.

Let S = {ui,u2,... ,un} be a finite set of vectors. Prove that S is linearly dependent if and only if ui = 0 or Wfc+i span({tii, w2 ,..., Uk}) for some k (1 < k < n).

Prove that a set S of vectors is linearly independent if and only if each finite subset of S is linearly independent.

Let M be a square upper triangular matrix (as defined in Exercise 12 of Section 1.3) with nonzero diagonal entries. Prove that the columns of M are linearly independent

Let S be a set of nonzero polynomials in P(F) such that no two have the same degree. Prove that S is linearly independent

Prove that if {Ai,A2, ,Ak} is a linearly independent subset of Mnxn(-F)j then {A\, A2, - , Al k} is also linearly independent

Let /, g, F(R, R) be the functions defined by f(t) = e rt and g(t) = e st , where r ^ s. Prove that / and g are linearly independent in J-(R, R)