Mathematics, 21.08.2019 05:10 lwattsstudent

Consider the real vector space v = r 4 . for each of the following five statements, provide either a proof or a counterexample. (a) dim v = 4. (b) span((1, 1, 0, , 1, 1, , 0, 1, 1)) = v . (c) the list ((1, −1, 0, , 1, −1, , 0, 1, −−1, 0, 0, 1)) is linearly independent. (d) every list of four vectors v1, . . , v4 ∈ v , such that span(v1, . . , v4) = v , is linearly independent. (页码61).

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Answer from: 10040816

Answer with Step-by-step explanation:

We are given that a real vector space

V= R^4

a.We have to prove that dim V =4

Let (a,b,c,d) is an element of V

Suppose that four elements

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

a(1,0,0,0)+b(0,1,0,0)+c(0,0,1,0)+d(0,0,0,1)

=(a,b,c,d)

All four elements are independent and can span each element of V.

Hence, dimension of V=4

b.We are given that (1,1,0,0),(0,1,1,0,(0,0,1,1)

We have to prove that these elements can span V

If number of elements are less than dimension of then the elements can not span V.

$\left[\begin{array}{cccc}1&1&0&0\\0&1&1&0\\0&0&1&1\end{array}\right]$

Rank of matrix =3

Hence, rank is less than the dimension of V.Therefore, given elements can not span V.

c.We are given that (1,-1,0,0),(0,1,-1,0),(0,0,1,-1),(-1,0,0,1)

We have to show that these elements are linearly independent.

$\left[\begin{array}{cccc}1&-1&0&0\\0&1&-1&0\\0&0&1&-1\\-1&0&0&1\end{array}\right]$

Every row or column is not a linear combination of other rows or columns.

Therefore, these elements are linearly independent.

d.We are given that $v_1,v_2,v_3,v_4\in V$ and span (v_1,v_2,v_3,v_4)=V

We have to prove that given vectors are linear independent.

Let v_1=(1,0,0,0),v_2=(0,1,0,0),v_3=(0,0,1,0),v_4=(0,0,0,1)

(a,b,c,d)=a(1,0,0,0)+b(0,1,0,0)+c(0,0,1,0)+d(0,0,0,1)

All four vectors are linearly independent because any element is not a linear combination other elements .

Hence, every four vectors $v_1,v_2,v_3,v_4\in V$ are linearly independent.

Answer from: Quest

The answer is 41.7 because 1.39 times 30 is 41.7

Answer from: Quest

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