Finding a basis in V, determining if a vector is in V, finding coordinates?

Question

Vector space V is spanned by vectors (2, -1, 2) and (1, -2, 2)
a. Find an orthogonal basis in V
b. Is vector v = (1, 4, 2) in V?
c. Find the coordinates of vector v relative to the basis you found in question "a"

Answer 1

In order to span a vector space of dimension 3 you need three vectors. You only list 2.

Answer 2

Vector space V is spanned by vectors
t = <2, -1, 2> and u = <1, -2, 2>.

a. Find an orthogonal basis in V

All the vectors in V or orthogonal to the normal vector n. Take the cross product.

n = t X u = <2, -1, 2> X <1, -2, 2> = <2, -2, -3>

Now find a vector in V that is orthogonal to both u and n. Take the cross product.

w = u X n = <10, 7, 2>

The orthogonal basis for V is t and w.
t = <2, -1, 2>
w = <10, 7, 2>
___________

b. Is vector v = <1, 4, 2> in V?

Solve for a and b.

at + bu = v

a<2, -1, 2> + b<1, -2, 2> = <1, 4, 2>

2a + b = 1
-a - 2b = 4
2a + 2b = 2

These equations lead to an inconsistency. There is no solution. So v does NOT lie in V.

Now that I think about it, you could also check to see if vector v lies in V. If it does, its cross product with the normal vector should equal zero. It does not. So again, v does NOT lie in V.

c. Not possible since v does not lie in V.

Answer 3

All you're able to desire to do is learn the 7 or 8 axioms indoors the definition of a vector section. as an celebration, between the axioms says that scalar multiplication ought to desire to distribute over vector addition: r(v + w) = rv + rw, the area r is a scalar and v and w are vectors. indoors the case of R as a vector section over Q, r is a rational determination and v and w are any truly numbers. yet then r,v,w are all truly numbers and the distributive ingredients of multiplication of top numbers is a properly many circumstances happening fact. extra desirable, the various axioms of a vector section additionally shop on with trivially from the worry-loose arithmetic properties of top numbers. as an celebration, 0 (in R) is the additive id of the vector section, and addition in R is the two associative and commutative.