Pfoof question please answer 1, 2, 3 Let S = span{u_1, u-2, ... u_m} (a) Show th

Question

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.

Explanation / Answer

Prrof 2).

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

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,.............um,v). and S1 = Span(u1,u2,.............um) .

let us first show S1 S0

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

Hence S1 S0.

Let us now show that S0 S1.

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

W = b0u + b1u1 + b2u2 + . . . + bmum.

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

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

w = b0(c1u1 + c2u2 + . . . + cmum) + b1u1 + b2u2 + . . . + bmum.

rearrange this expression

w = (b0c1 + b1)u1 + (b0c2 + b2)u2 + . . . + (b0cm + bm)um.

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

Hence Proved that these 2 sets are equal.

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