Linear Transformations: One-to-One and Onto Conditions

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SUMMARY

The discussion focuses on the conditions for linear transformations T: R3 -> R5 being one-to-one and onto. For a one-to-one transformation, the only valid conclusion is that its rank must be three and its nullity must be two, as it cannot exceed the dimension of the domain. Conversely, for an onto transformation, the rank must be five, which is impossible given the dimensional constraints of the transformation. Thus, the correct answers are that a one-to-one transformation has a rank of three and nullity of two, while an onto transformation is impossible in this context.

PREREQUISITES
  • Understanding of linear transformations and their properties
  • Knowledge of rank and nullity in linear algebra
  • Familiarity with the concepts of one-to-one and onto mappings
  • Basic comprehension of vector spaces and dimensions
NEXT STEPS
  • Study the Rank-Nullity Theorem in linear algebra
  • Explore examples of linear transformations between different dimensions
  • Learn about the implications of one-to-one and onto transformations in Rn
  • Investigate the concept of null space and its significance in linear transformations
USEFUL FOR

Students studying linear algebra, particularly those focusing on linear transformations, rank, and nullity. This discussion is beneficial for anyone preparing for exams or needing clarification on the properties of linear mappings.

phrygian
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Homework Statement




(124) If a linear transformation T : R3 -> R5 is one-to-one, then
(a) Its rank is five and its nullity is two.
(b) Its rank and nullity can be any pair of non-negative numbers that add
up to five.
(c) Its rank is three and its nullity is two.
(d) Its rank is two and its nullity is three.
(e) Its rank is three and its nullity is zero.
(f) Its rank and nullity can be any pair of non-negative numbers that add
up to three.
(g) The situation is impossible.

(125) If a linear transformation T : R3 -> R5 is onto, then
(a) Its rank is five and its nullity is two.
(b) Its rank is two and its nullity is three.
(c) Its rank is three and its nullity is zero.
(d) Its rank and nullity can be any pair of non-negative numbers that add
up to three.
(e) Its rank is three and its nullity is two.
(f) Its rank and nullity can be any pair of non-negative numbers that add
up to five.
(g) The situation is impossible.








Homework Equations





The Attempt at a Solution




These two problems on my practice test have me completely stumped, could someone help shed some light?

I understand the definitions of onto and one-to-one, but don't understnad how to connect this to the null space of a linear transformation from Rm to Rn?
 
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Saying that a linear transformation is "one-to-one" means that only one "u" is mapped to a specific "v", right? In particular, that means only one vector is mapped to the 0 vector. Since any linear transformation maps the 0 vector to the 0 vector, a one to one mapping maps only the 0 vector to the 0 vector.

That is, a linear transformation is one to one if and only if its nullspace is the "trivial" space consisting only of the 0 vector.
 

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