Please help me with the following Linear Algabra questions P

Please help me with the following Linear Algabra questions. Please show work and write neat.

Show that the set is linearly dependent by finding a nontrivial linear combination of vectors in the set whose sum is the zero vector. (Use s_1, S_2, and s_3, respectively, for the vectors in the set.) S = {(1, 2, 3, 4), (1, 0, 1, 2), (1, 4, 5, (0, 0, 0. 0) = Express the vector S_3 in the set as a linear combination of the vectors s_1 and S_2.s_3 = What steps or reasoning did you use? Your work counts towards your score You can submit show my work an unlimited number of times Show My Work has not been graded yet Write the standard basis for the vector space.

Solution

Let s₁ = (1,2,3,4)^T, s₂ = (1,0,1,2)^T and s₃ = (1,4,5,6)^T. Also, let A =

1

1

1

2

0

4

3

1

5

4

2

6

We will reduce A to its RREF as under:

Add -2 times the 1st row to the 2nd row

Add -3 times the 1st row to the 3rd row

Add -4 times the 1st row to the 4th row

Multiply the 2nd row by -1/2

Add 2 times the 2nd row to the 3rd row

Add 2 times the 2nd row to the 4th row

Add -1 times the 2nd row to the 1st row

Then the RREF of A is

1

0

2

0

1

-1

0

0

0

0

0

0

Now, it is apparent that s₃ =2s₁-s₂ and s₃-2s₁+s₂ =0.

4. A standard basis for the vectoor space M_4,1 is { (1,0,0,0)^T,(0,1,0,0)^T, (0,0,0,1)^T, (0,0,0,1)^T}

1	1	1
2	0	4
3	1	5
4	2	6