I believe the first 3 are true but still unsure about all of

I believe the first 3 are true but still unsure about all of them. There are two other postings on Chegg that have them all true which is not correct.

(1 point) Are the following statements true or false? True 1 . For any scalar c, u\"(cv) = c(uTv). True 2. Let u and v be non zero vectors. If the distance from u to v is equal to the distance from u to -v, then u and v are orthogonal True 3. If vectors vi, . . . ,y, span a subspace W and if x is orthogonal to each vi for j = L . ..,p. then x Is in wi. ? \' 4. For a square matrix A, vectors in R(A) are orthogonal to vectors in N(A).

Solution

1. u^T(cv) = c(u^Tv)

It is true.

2. let a = (3 , 5) and b = (5, -3) ; -b = (-5 , 3)

|b -a| = sqrt( 2^2 + 8^2)

|-b -a| = sqrt(8^2 +2^2)

a and b are orthogonal

u and v are orthogonal.True

5) v^Tv = ||v||^2

Let v = ix +jy ; v^T = ix -jy

v^Tv = (ix +jy)(ix -jy = (x^2 - y^2 )

So , it is not equal to ||v||^2

False