If the following statement is true, give a proof: if it is false, show it with a
ID: 1888156 • Letter: I
Question
If the following statement is true, give a proof: if it is false, show it with an example. If a linear transformation on vector spaces T : V rightarrow W is injective, then the image of a linearly independent set in V is a linearly independent set. in W. Prove the following statement.: A linear operator T : V rightarrow V on a finite dimensional vector space V is injective iff T is surjective. Prove the following statement: If V and W are finite dimensional vector spaces with dim V > dim W then every linear transformation T: V rightarrow W is not. injective.Explanation / Answer
Since T:V-->W is injective for every linearly independent vector ( basis ) in the Domain we have a linearly independent basis in the Range hence The statement is true. If T is not surjective then we can have one or more elements in V mapped to element in range of V which contradicts V being surjective hence the mapping is injective. Give dim V > dim W Hence the number of elements in basis V is greater than that of W. Hence there are more elements in V than in W hence there need not be a one - one relation.