Vectors, Linear Map; Injective and surjective

Question

Let vector, v = - j + k in R³

f : R³ ---> R³

f(u) = v x u + 2u for u in R³

(u, v, j, k are vectors)

How can we determine if f is surjective or not?

How can we determine if f is injective or not?

Answer 1

Surjective is onto. I cross product any u with the given vector and add u.

I wind up with a matrix times u. Equate that product to a vector k which

is any vector in real space. If I u₁,u₂,u₃ can be soved for than the exprssion is surjective. This means the matrix determinant must not be zero.

the rows of the matrix I computed are:(1,-1,-1);(-1,1,0),;(-1,0,1)

Its determinant is not zero so expression is surjective.

injective means one to one releationship between any single u and the expression outcome.

set expression = to two vectors, l₁ and l₂

show if l₁ ≠ l₂ then v cross u is unequal in some circumstance to v cross u ;;;;;;;;;;;

a contradiction so the expression is injective