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Let [tex]T:\mathbb{R}^n \to \mathbb{R}^m[/tex] be a linear transformation with standard matrix [tex]A[/tex]. Which of the following is/are true? (Select all that apply)

A. If [tex]T[/tex] is onto, the columns of [tex]A[/tex] are linearly independent.
B. If [tex]T[/tex] is one-to-one, [tex]Ax=0[/tex] has a nontrivial solution.
C. If [tex]T[/tex] is onto, [tex]Ax=b[/tex] has at least one solution for every [tex]b \in \mathbb{R}^m[/tex].
D. If [tex]T[/tex] is one-to-one, [tex]A[/tex] has a pivot in every row.
E. If [tex]T[/tex] is onto, the range of [tex]T[/tex] is [tex]\mathbb{R}^m[/tex].

Answer :

Final answer:

The correct statements about properties of a linear transformation with reference to its standard matrix A are C.If T is onto, Ax=b has at least one solution for every b in Rm and E. If T is onto, the range of T is Rm.

Explanation:

The subject of the question pertains to properties of linear transformation, particularly in terms of the matrix representation of the transformation and its characteristics. Let's go through each statement one by one:

  • A. If T is onto, it's not necessarily true that the columns of A are linearly independent. For a transformation to be onto, it indicates that every element in the range is mapped to from the domain, but it doesn't necessarily mean that columns of A (vectors in domain) are linearly independent.
  • B. If T is one-to-one, Ax=0 does not have a nontrivial solution. By definition, a one-to-one transformation implies that only the null vector in the domain gets mapped to the null vector in the codomain, meaning that Ax=0 only has the trivial solution.
  • C. If T is onto, Ax=b does, indeed, have at least one solution for every b in Rm. This statement is a restatement of the definition of an onto function.
  • D. If T is one-to-one, A does not necessarily have a pivot in every row; however, A will have a pivot in every column.
  • E. If T is onto, the range of T is Rm. This definition is accurate for onto transformations.

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