Linear Algebra Suppose T : Rn Rm is a linear transformation and let A be its ass

Question

Linear Algebra

Suppose T : Rn Rm is a linear transformation and let A be its asso- ciated matrix, so that T(v) = Av for all vectors v Rn. Recall that the column space of A, denoted by Col(A) is the subspace of R" spanned by the columns of A. It is the same as the range of T, denoted by Range(T), which is the subspace of all of the vectors of Rm that "get hit" by elements of the domain. The nullspace of A, denoted by Null(A) is the same as the nullspace of T, denoted by Null(T). It is the subspace of R" consisting of all vectors that get mapped to 0. Answer the following questions. You must explain what you are doing and why you are doing it

Explanation / Answer

a)

v_3=-3v_2-2v_1

Hence, v1,v2,v3 are linearly dependent

But, v_1 and v_2 are linearly independent

Hence, null(T)=span{v1,v2{

dimension of Null(T)=2

b)

Basis for Null(T)={v1,v2}

c)

By Rank nullity theorem

dim range(T)+dim Null(T)=5

Hence dimension of range(T)=3

But range(T) is subset of R3 hence, range(T)=R3

So the basis for Range(T)=standard basis for R3

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