Linear Transformations, Null Spaces & Ranges
8 important questions on Linear Transformations, Null Spaces & Ranges
What is the Null Space/ Kernel?
-V & W are vector spaces, T: V->W is linear, then nullspace of T N(T) is the set of all vectors (x) in V, such that T(x)= 0
-N(T) : { x in V : T(x)=0}
- all vectors x in V that become 0 vectors in W after applying transformation T
-Subspace of V
What is the Range/ Image?
-V & W are vector spaces, T: V-> W is linear, then Range (R(T)) is the subset of W containing all the images of vectors x in V (T(x)) under transformation T
- R(T) : { T(x) : x in V}
- all vectors T(x) in W that become x vectors in V if we "cancel" transformation T
-Subspace of W
What is the Zero Transformation?
-V & W are vector spaces of F, then zero transformation is denoted by To, (from V to W), To: V->W
- To(x) = 0 for all x in V
-every vector x in V to which we apply the zero transformation To becomes zero vector 0 (in W).
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How to find a Spanning Set for the Range of a Linear Transformation? (Theorem 2.2)
T: V->W is linear, and
B={v1, v2,...,vn} is basis for V.
Then, R(T)= Span(T(B))= Span{ T(v1), T(v2),..., T(vn)}
- meaning that R(T) is contained in the Span(T(B))
- this is true when B is infinite that is R(T)= Span ({T(v): v in B})
What are the dimensions of Null Space and Range called?
And N(T) and R(T) are finite dimensional, then we denote nullity(T) as the dimension of null space and rank(T) as the dimension of the range
What is the Dimension Theorem?
it is proven because if V and W are vector spaces and T:V->W is linear, and V is finite dimensional,
then nullity(T) + rank(T) = dim(V)
What does it mean for a linear transformation to be one-to-one?
- T:V->W is 1-1 if the equation T(x)=b has at at most one solution for every b
(looking from x to y)
What can be concluded from a linear transformation of finite dimensional vector spaces of equal dimensions?
T being 1-1 is equivalent to T being onto and it's equivalent to rank(T)= dim(V)
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