Range in Linear Transformation

[mrshappy0]

Homework Statement

L: R^3 -> R^2
L(x)=(0,0)^T
What is the basis, and dim of the Range?

Homework Equations

Rank(A)-Nullity(A)=n

The Attempt at a Solution

So clearly L(x)= (0,0)^T. So the basis must be the empty space and dim is zero, right?

Now, going of this same logic, Say L(x)=(x2,x3)^T. The basis would be {(1,0)^T, (0,1)^T} does this mean the range is just the Span of these two linearly independent vectors--> Span (v1,v2)?

 

[Mark44]

Homework Statement

L: R^3 -> R^2
L(x)=(0,0)^T
What is the basis, and dim of the Range?

Homework Equations

Rank(A)-Nullity(A)=n

The Attempt at a Solution

So clearly L(x)= (0,0)^T. So the basis must be the empty space and dim is zero, right?

Every vector space must consist of at least the zero vector, so a basis for the range of L would be <0, 0>. The dimension of the range is zero, which is what you said.

Now, going of this same logic, Say L(x)=(x2,x3)^T. The basis would be {(1,0)^T, (0,1)^T} does this mean the range is just the Span of these two linearly independent vectors--> Span (v1,v2)?

That's what a basis is - a set of vectors that spans some space or subspace. In this example, the range is all of R².

 

[mrshappy0]

#44, for the win. Thanks you.