Pfoof question please answer 1 2 3 Let S spanu1 u2 um a Sh

Pfoof question

please answer 1, 2, 3

Let S = span{u_1, u-2, ... u_m} (a) Show that if upsilon S, and is a scalar, then cv S. (b) Show that if upsilon S and w S. Suppose that upsilon span {u_1, u_2, ...u_m}. Show that span {u_1, u_2, ...u_m} = span {u_1, u_2, ...u_m, upsilon}. Consider the following statements about vectors u_1, ..., u_m, upsilon in R^n. (i) {u_1, ..., u_m} is linearly independent in R^n. (ii) upsilon in R^n does not lie in span {u_1, ..., u_m}. (iii) {u_1, ..., u_m, upsilon} is linearly independent. Answer the following questions. (a) Fidn an example of u_1, ...u_m, upsilon so that (i) is true, (ii) is false, and (iii) is false. (b) Find an example of u_1, ...u_m, upsilon so that (i) is false, (ii0 is true, and (iii0 is false. (c) Show that if (i0 and (ii0 are true, then (iii0 is true.

Solution

Prrof 2).

Basically we need to prove that V is redundant in Span(u₁,u₂,.............u_m).

If we want to show equality of two sets, we do that by showing that one set is contained in the other and vice versa.

Let S₀= Span(u₁,u₂,.............u_m,v). and S₁ = Span(u₁,u₂,.............u_m) .

let us first show S₁ S₀

Let b be any element in S₁.
Then there are scalars a₁, a₂, . . . , a_m, such that b = a₁u₁ + a₂u₂ + . . . + a_mu_m.
This sum can also be written as b=  0v + a₁u₁ + a₂u₂ + . . . + a_mu_m.
so that b lies in S₀.

Hence S₁ S_0.

Let us now show that S₀ S₁.

Fix W S₀, so there are scalars, b₀, b₁, b₂ . . . , b_m such that

W = b₀u + b₁u₁ + b₂u₂ + . . . + b_mu_m.

Assume u is an elememt in S₁.That means, that there are scalars c₁, c₂, . . . , c_m such that
u = c₁u₁ + c₂u₂ + . . . + c_mu_m.

Let us plug in this representation of u in the representation of W. We then get

w = b₀(c₁u₁ + c₂u₂ + . . . + c_mu_m) + b₁u₁ + b₂u₂ + . . . + b_mu_m.

rearrange this expression

w = (b₀c₁ + b₁)u₁ + (b₀c₂ + b₂)u₂ + . . . + (b₀c_m + b_m)u_m.

This is a linear combination of the vectors {u₁, u₂, . . . , u_m}, hence it lies in S1.

Hence Proved that these 2 sets are equal.