In Exercises 19 and 20 all vectors are in R Mark each statem

In Exercises 19 and 20, all vectors are in R\" Mark each statement True or False. Justify each answer A is in b. For any scalar c llcvill 3cllvl If x is orthogonal to every vector in a subspace w then x is in WL If u llvl? u t v then u and v are orthogonal For an m x A. vectors in the null space of A are orthogonal to vectors in the row space of A

Solution

1) The first statement is TRUE, since the dot product is commutative i.e. u.v = v.u

since u.v = |u||v|cos(theta)

and v.u = |v||u|cos(theta)

which implies the first statement is true

2) The given statement is FALSE, for any value of c<0, since the norm of vector is always a positive quantity hence there must be a case when c<0. which makes the left hand side positive and right hand side value negative

The correct formula is ||cv|| = |c|*||v||

3) The given statement is TRUE. since the vector x is orthogonal to W implies

x.W = 0

which implies all vector lies in the orthognal space of W

4) The given statement is TRUE, since if the vectors are orthogonal then u.v will be equal to zero

||u+v||^2 = ||u||^2 + ||v||^2 + 2u.v*cos(theta)

since u.v =0, hence the ||u+v||^2 = ||u||^2 + ||v|^2

5) The given statement is TRUE, since the dot product of row space and null space is always zero