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AC= (˜1 vz вЂ˜.. vk) ;

0

Ck

has a nonzero solution c. But by Fact 7.6 in Section 7.4, any matrix A with more

columns than rows will have a free variable and therefore AC = 0 will have

infinitely many solutions, all but one of which are nonzero. n

EXERCISES

11.1 Show that if (1) or (2) holds, then (3) holds and, if (3) holds, then (1) or (2) holds.

11.2 Which of the following pairs or triplets of vectors are linearly independent?

a) (2, I), (1,2); b) (2, 1)) (-4, -2);

4 (4 LO), (4 1, I), (LO, I),

4 (1, 1, fl), (03 1. 1);

11.3 Determine whether 01 not each of the following collections of vectors in RвЂ™ are

linearly independent:

11.4 Prove that if (4) holds, then v, is not a multiple of v1 and vz is not a multiple of vi.

LINEAR INDEPENDENCE [1 11

244

11.5 a) Show that if vl, ˜2, and vi do not satisfy (5), they satisfy (6), and vice versa.

b) Show that (5) is equivalent to the statement that one of v,, v2, and v3 is a linear

combination of the other two.

1 1 . 6 Prove that any collection of vectors that includes the zero-vector cannot be linearly

independent.

11.7 Prove Theorem 11.1.

11.8 Prove Theorem 11.2.

11.2 SPANNING SETS

Let vI, , vk be a fixed set ofk vectors in RвЂќ. In the last section, we spoke of the

set of all linear combinations of ˜1, , YX,

L[v,, , vk] = {qv, + + ckvx : cl, , ck E R},

and called it the set generated or spanned by vI,. , vk.

Suppose that we are given a subset V of RвЂќ. It is reasonable to ask whether

or not there exists v,, , vk in RвЂќ such that every vector in V can be written as a

linear combination of v,, , vk:

v = L[v,, , Vk]. (10)

When (10) occurs, we say that vI, , vq s&mвЂќ V.

Every line t h r o u g h t h e origin is the span of a nonzero vector on the

Exumpk 11.3

For example, the x,-axis is the span of e, = (I, 0,. , 0). and the diagonal

A = {(u, a.. a) t RвЂќ : a E R}

is the span of the vcctw (I, 1.. , 1).

Example II.1 The r,xz-plane in RJ is the span of the unit vectors e, = (LO, 0)

and e: = (0, I, 0), because any vector (a, h, 0) in this plane can be written as

Exumple11.5 The edimensional Euclidean space itself is spanned by the vectors

el, , e, of Example 11.1. For, if (N,, , aI?) is an arbitrary vector in RвЂќ, then

Ill .21 SPANNING SETS 245

we can write

Example II.6 Different sets of vectors can span the same space. For example,

each of the following sets of vectors spans RвЂ™:

0) (A), (3

4 (-A), (:)I

4 (:j, (;)a (:);

I

4 (-i). (:):

I

p) (-I): ($ (i).

Theorem 1 I ,l presented a matrix criterion for checking whether a given ret of

sectors is linearly independent. The following theorem carries out the analogous

task for checking whether a set of vectors s,@ans.

Theorem 11.4 Let vl, , vi he a set of k vectors in RвЂќ. Form the n X k

matrix whose columns are these viвЂ™s:

A=(v, вЂњ2 .вЂ˜. ˜4). (11)

Let b be a vector in RвЂќ. Then. h lies in the space r[v,. , VK] spanned by

v,, vi if and only if the system AC = b has a solution c.

LINEAR INDEPENDENCE I1 1 I

246

Then, b is in I[v,, , vk] if and only if we can find ct,. , ck such that

clвЂќ, + + ckvk = b,

or (12)

1So, b E I[v,, _, vx] if and only if system (12) has a solution c. n

The following corollary of Theorem 11.4 provides a simple criterion for whether

or not a given set of vectors spans all of RвЂќ. Its proof is left as a simple exercise.

Theorem 11.5 Let v,. , vk be a collection of vectors in RвЂќ. Form the n X k

matrix A whose columns are these v,вЂ˜s, as in (11). Then, v,, , vk span RвЂќ if

= b has a solution x for every right-hand

and only if the system of equationsAx

side b.

in Example 11.5. we found n vectors that span RвЂќ. In Example 11.6, we listed

various collections of two or three vecton that span RвЂ™. Clearly, it takes at least

two vectors to span RвЂ™. The next theorem, which follows easily from Theorem

11.5, states that one needs at least n vectors to span RвЂќ.

Theorem 11.6 A set of vectors that spans RвЂќ must contain at least n vectors.

Proof By Theorem 11.5, vI,. , vx span RвЂќ if and only if system (12) has a

solution E for every right-hand side b E RвЂќ. Fact 7.7 tells us that if system (12)

has a solution for each right-hand side, then the rank of the coefficient matrix

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