Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 3198. Show that the set of all possible solutions (x, y, z) ∈ R 3 of the equationsforms a subspace of R 3 .Ax + By + Cz = 0Dx + Ey + Cz = 09. Let W be the subset of continuous functions on [0, 1] such that f(0) = 0.Prove that W is a subspace of C[0, 1].10. Let V be a vector space over F . Prove that −(αv) = (−α)v = α(−v) for allα ∈ F and all v ∈ V .11. Let V be a vector space of dimension n. Prove each of the following statements.(a) If S = {v 1 , . . . , v n } is a set of linearly independent vectors for V , thenS is a basis for V .(b) If S = {v 1 , . . . , v n } spans V , then S is a basis for V .(c) If S = {v 1 , . . . , v k } is a set of linearly independent vectors for V withk < n, then there exist vectors v k+1 , . . . , v n such thatis a basis for V .{v 1 , . . . , v k , v k+1 , . . . , v n }12. Prove that any set of vectors containing 0 is linearly dependent.13. Let V be a vector space. Show that {0} is a subspace of V of dimension zero.14. If a vector space V is spanned by n vectors, show that any set of m vectorsin V must be linearly dependent for m > n.15. Linear Transformations. Let V and W be vector spaces over a field F , ofdimensions m and n, respectively. If T : V → W is a map satisfyingT (u + v) = T (u) + T (v)T (αv) = αT (v)for all α ∈ F and all u, v ∈ V , then T is called a linear transformationfrom V into W .(a) Prove that the kernel of T , ker(T ) = {v ∈ V : T (v) = 0}, is asubspace of V . The kernel of T is sometimes called the null space ofT .(b) Prove that the range or range space of T , R(V ) = {w ∈ W :T (v) = w for some v ∈ V }, is a subspace of W .(c) Show that T : V → W is injective if and only if ker(T ) = {0}.