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.mrshappy0 said: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?
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 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)?
The range in linear transformation refers to the set of all possible outputs or values that can be obtained from a linear transformation. It is also known as the image or codomain of the transformation.
The range of a linear transformation can be determined by applying the transformation to all possible inputs and observing the resulting outputs. The set of all these outputs will form the range of the transformation.
The range is important in linear transformation because it gives us information about the span and dimension of the transformation. It also helps us understand the behavior and properties of the transformation.
No, the range and domain of a linear transformation are usually different. The domain refers to the set of all possible inputs, while the range refers to the set of all possible outputs. In most cases, the range will be a subset of the domain.
The range of a linear transformation can be expanded by using a different basis or changing the dimension of the domain. Additionally, the range can also be expanded by composing the transformation with other linear transformations.