Please show answers clearly, thanks. Show that the set of vectors is linearly in

Question

Please show answers clearly, thanks.

Show that the set of vectors is linearly independent and spans R3. B= {(1, 2, 3), (3, 2, 1), (0 ,0, 1)} Explain why each set below, S, is not a basis for the given vector space. S = {(2, 3), (6, 9)} V = R2 s = {(1, 1, 2),(0,2, 1)} V = R2 S = {0, x,3 x2} V = P2 V = M2,2 Consider the following subspace of R3 : W = {(2s - t, s, t): s and t are real numbers}. Find a basis for W. Determine the dimension of W. Give a geometric description of the subspace. Given that A and B are row equivalent, find the following: the rank and nullity of A, a basis for the nullspace of A, a basis for the row space of A, a basis for the column space of A, and the columns of A that are linearly independent. Let A be an m x n matrix (where m

Explanation / Answer

1.

(a)

let

a(1,2,3)+b(3,2,1)+c(0,0,1) = 0

=>
a+3b = 0

2a+2b = 0

3a+2b+c = 0

=>
a = 0, b = 0, c = 0

=>

they are linealry independent

let (x,y,z) = a(1,2,3)+b(3,2,1)+c(0,0,1)

=>

a+3b = x

2a+2b = y

3a+2b+c = z

=>

b = (2x-y)/4, a = (y-x)/2, c = z-3(y-x)/2 - 2(2x-y)/4

=>

they span R^3

2.

(a)

(1,0) is not in the span of the given set

(b)

let (1,1,1) = a(1,1,2)+b(0,2,1)

=>

1 = a, 1 = a+2b, 1 = 2a+b

which is not possible

=>

(1,1,1) is not in the span of the given set

(c)

{1} is not in the span of the given set

3.

(a)

basis = {(2,1,0),(-1,0,1)}

(b)

dimension = 2

(c)

plane

4.

(a) rank(A) = 4, nullity (A) = 1

(b)rank(B) = 3, nullity (B) = 2

5.

(a)m

(b)r

(c)r

(d)R^r

(e)R^r

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