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Practice Problems

(1) Determine if the vector (-2,0,3), can be expressed as a linear combi-nation of the vectors (1,3,0) and (2,4,-1).Solution: (2, 3, 0) = 4(1, 3, 0) + (3)(2, 4,1).

(2) Determine if the polynomial 2x3 2x2 + 12x 6 can be expressedas a linear combination of x3 2x2 5x 3 and 3x3 5x2 4x 9.Solution: 2x3 2x2 + 12x 6 = (4)(x3 2x2 5x 3)+2(3x35x2 4x 9). (of course, you have to find a solution of a suitablesystem of equations to come up with this answer!

(3) Determine if the polynomial 3x3 2x2 + 7x + 8 can be expressed asa linear combination of x3 2x2 5x 3 and 3x3 5x2 4x 9.Solution: Start with 3x3 5x2 4x 9 = a(x3 2x2 5x 3) +b(3x3 5x2 4x 9) and try to find a s olution of the system of

equations. Show that there are no such a and b.(4) Determine if the matrix

1 23 4

is in the span of

S = {A =

1 01 4

, B =

0 10 1

, C =

1 10 0

}.

Solution: Note that 3A + 4B2C =

1 23 16

. Thus, the given

matrix is not in the span of A,B,C.(5) Prove that span({x}) = {ax|a F}.

Solution: Follows from the definition.(6) Show that a subset W of a vector space V is a subspace of V if and

only if span W = W.Solution: If span W = W then W contanis all linear combinationsof its elements. That is , if x, y W then (i) x + y,, (ii) 0 W, and (iii) x W for F. Hence W is a subspace of V.Conversely, if W is a subspace of V then for any subset {u1,....,un}in W, 1u1 + .... + nun W for every 1, .....n F. (why?) Thus,spanW W. Hence the result.

(7) Show that ifS1, S2 are subsets of a vector space V such that S1 S2,then span(S1) span (S2). If S1 S2 and span(S1) = V, deducethat span(S2) = V.Solution: If Let u spanS1. Then, u =

m

i=1iui for ui S1,

i F. But, if ui S1 then ui S2. Thus The RHS is also inspan S2. That implies, U span S2. To prove the second part, startwith v V. Since, S1 S2, from the first part it follows spanS1 spanS2. This means V span S2. Hence, V =span S2.

(8) Show that S1 and S2 are arbitrary subsets of a vector space V thenspan(S1 S2) = span(S1) + span(S2).Solution: Let u span (S1 S2), then

u =

mi=1

iui +

ni=1

ivi

for ui S1, i = 1,..m and vi S2, i = 1, ...n, and i, i F. Then,

u spanS1 + spanS2.1

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(9) Let S1 and S2 be subsets of a vector space V. Prove that (S1S2) span(S1) span(S2). Give an example in which span(S1 S2) and

span(S1) span(S2) are are equal and one in which they are notequal.Solution: Start with an element in S1S2 and use the fact that Sspan S, for a subset S of V. When S1 = S2 clearly, span(S1 S2) =span S1 span S2. ( For an explicit example, start with two setseach containing a single (different) element of IR2. Check the result.Let S1 = {(1, 0), (0, 1)} and S2 = {(1, 1)}. Span S1 = IR

2. spanS2 ={(x, y)|y = x}; clearly span S1span S2 = span S2. Since, S1S2 = span S1 S2 = {0}.