1By inspection determine ii the columns of A given below for

1.By inspection determine ii the columns of A given below form a Linear Independent set and provide your explanation 2.) The given Augmented matrix of a system is in reduced echelon form. Write the solution of the system in vector form. 4.) Elementary row operation co not change the solution of the original system of linear equations...Why is this true? S.) A set of vectors is said to be Linearly Independent if the corresponding system of homogenous equations admits more than one solution...Why is this false?

Solution

Feel free to ask if you have any queries.

1) They are dependent.

4 vectors of R³ will be always be dependent. (Dimension of R3 is 3)

The reason is you can always write any vector as linear addition of basis vectors.

Since dimension is 3, 4th vector can always be written in form of addition of 3 independent vectors,

and thus the set of 4 vectors will be dependent.

2)

Solution is x_p + x_H (x_H is homogenous solution)

where x_p is [3 6 0 0 0]\'    (by setting free variables to 0.)

x_H is x3*[8 -7 1 0 0]\' + x4*[0 1 0 1 0]\' (by using the identity that homogenous solution is [-F I] appended by [O O])

4)

Swapping rows is just changing the order of the equations begin considered, which certainly should not alter the solutions. Scalar multiplication is just multiplying the equation by the same number on both sides, which does not change the solution(s) of the equation. Likewise, if two equations share a common solution, adding one to the other preserves the solution.

I can also provide rigorous mathematics proof but what matters ithe conceptual insight in this case.

5) The definition of independent vector is that the solution to any linear addition of vectors is zero if and only if the coefficient of each vector is zero.

Therefore solution to homogenous equation is only one and that is all 0s.