Please provide the process for solution. solution to each problem must show all

Question

Please provide the process for solution.

solution to each problem must show all major steps, and be organized.

13. W is a subset of R2 defined as

W ={(x,y)|x?0, y?0}.

(a) Is W closed under vector addition? If your answer is no,then find two vectors u,v?W such that u+v??W (not included)

(b) Is W closed under scalar multiplication?
If your answer is no, then find a vector u ? W and a scalar c such that cu ?? W(not included).

(c) Is W a subspace of R2?

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14. Let

v1 = (1,1,1,1), v2 = (1,2,1,4), v3 = (1,0,1,0).

Determine whether the vectors v1, v2, v3 are linearly independent.

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15. Let

v1 =(1,2,3), v2 =(3,2,1), v3 =(1,1,1).

Determine whether the vectors v1, v2, v3 are linearly independent.

___________________________________________

16. Let

v1 =(1,2,3), v2 =(1,1,1), v3 =(1,0,1), v4 =(1,3,5).

  Determine whether the vectors v1, v2, v3, v4 span R3.

Explanation / Answer

For answers use these conditions....

If v1, v2, v3, v4 are linearly independent vectors in R^4, then {v1, v2, v3} must also be linearly independent.

And

If v1, v2, v3, v4 are in R^4 and v3 = 2v1 + v2, then {v1, v2, v3, v4} must be linearly dependent

and

If v1, v2, v3, v4 are in R^4 and v3 = 0, then {v1, v2, v3, v4} must be linearly dependent.

and

If v1, v2, v3, v4 are in R^4 and {v1, v2, v3}is linearly dependent, then {v1, v2, v3, v4} must also be linearly dependent

If {v1, v2, v3}were dependent, then there would be an equation if x1v1 + x2v2 + x3v3 = 0 with x1, x2, x3 not all zero. But then x1v1 + x2v2 + x3v3 + 0 v4 = 0 and still the coefficients are notALL zero so this contradicts the fact that {v1, v2, v3, v4} is linearly independent. Note: Questions 4 and 5 are logically equivalent. so  If v1, v2, v3, v4 are linearly independent vectors in R^4, then {v1, v2, v3} must also be linearly independent.

For span ::::

span of v1 = (1, 1) and v2 = (2, ?1) in R2?
Answer: R2

For R3 check Ax =v and if you can get the solution then it spans..

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