Describe the span of M = {(1, 1, 1), (0, 0, 2)} in R^3. Which of the following s

Question

Describe the span of M = {(1, 1, 1), (0, 0, 2)} in R^3. Which of the following subsets of R^3 constitute a subspace of R^3? [Here, x = (xi_1, xi_2, xi_3).] All x with xi_1 = xi_2 and xi_3 = 0. All x with x_1 = xi_2 + 1. All x with positive xi_1, xi_2, xi_3. All x with xi_1 - xi_2 + xi_3 = k = const. Show that {x_1, .. ., x_n}, where x_i(t) = t^j, is a linearly independent set in the space C[a, b]. Show that in an n -dimensional vector space X, the representation of any x as a linear combination of given basis vectors e_1, .. ., e_n is unique.

Explanation / Answer

a)

Let (a,b,c) and (e,f,g) be in the set

Then,

(a,b,c)+(e,f,g)=(a+e,b+f,c+g)

a+e=b+f

c+g=0

Hence set is closed under addition

Let, k be any scalar

So,

since a=b then, k*a=k*b

and , k*c=0

Hence set is closed under scalar multiplication

Hence a subspace

b)

Not a subspace

Consider two points in this set

(1,0,1) and (2,1,0)

Adding gives

(3,1,1) which is not in the set

Hence not closed under addition and hence not a subspace

c)

Not a subspace

Let, (1,1,1) be in the set

So, (-1)*(1,1,1)=(-1,-1,-1) is not in the set

d)

Depends on the value of k

For k=0 it is a subspace

For k non zero it is not

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