Linear Transformation and Linear dependence - Proof

In summary, we are given a linear transformation T: R^n -> R^m that maps two linearly independent vectors {u,v} to a linearly dependent set {t(u),T(v)}. To show that the equation T(x) = 0 has a nontrivial solution, we can use the fact that T(c1u1 + c2v2) = T(0) where c1 or c2 /= 0, and by substituting in c1T(u1) + c2T(v2) = 0 (c1 or c2 /= 0), we can see that T(x) = 0 has a trivial solution with c1T(x) = 0 where c1 /=
  • #1
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Homework Statement


Let T:Rn to Rm be a linear transformation that maps two linearly independents vectors {u,v} into a linearly dependent set {t(u),T(v)}. Show that the equation T(x)=0 has a nontrivial solution.

Homework Equations



c1u1 + c2v2 = 0 where c1,c2 = 0

T(c1u1 + c2v2) = T(0) where c1 or c2 /= 0

The Attempt at a Solution



Since we know:

T(c1u1 + c2v2) = T(0) where c1 or c2 /= 0
T(c1u1) + T(c2v2) = T(0)
c1T(u1) + c2T(v2) = T(0)
c1T(u1) + c2T(v2) = 0 (c1 or c2 /= 0)

T(x) = 0 has trivial solution
c1T(x) = 0 where c1 /= 0

I'm not sure how to connect those two ideas or if there even relevant to the solution proof.
 
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  • #2
I'm not sure what your last expression is supposed to mean, but the block of four equations is already the proof. Maybe it is easier to see it, if you read it backwards.
 

Related to Linear Transformation and Linear dependence - Proof

1. What is a linear transformation?

A linear transformation is a mathematical function that maps one vector space to another, while preserving the algebraic structure of the vectors. In other words, it is a function that takes in vectors as inputs and produces vectors as outputs, where the operations of addition and scalar multiplication are preserved.

2. How do you prove that a transformation is linear?

To prove that a transformation is linear, you must show that it satisfies two properties: additivity and homogeneity. Additivity means that the transformation of the sum of two vectors is equal to the sum of the individual transformations. Homogeneity means that the transformation of a scaled vector is equal to the scaled transformation of the original vector. If a transformation satisfies both of these properties, it is considered linear.

3. What is the difference between linear independence and linear dependence?

Linear independence refers to a set of vectors that cannot be created by linear combinations of each other. In other words, no vector in the set can be written as a linear combination of the other vectors. Linear dependence, on the other hand, refers to a set of vectors where at least one vector can be written as a linear combination of the others. This means that the vectors in a linearly dependent set are not all necessary to span the vector space.

4. How do you prove linear dependence?

To prove linear dependence, you must show that there exists a non-trivial linear combination of the vectors in the set that equals the zero vector. In other words, you must find coefficients for each vector such that when you multiply each vector by its corresponding coefficient, the resulting sum is equal to the zero vector. If such coefficients exist, the set of vectors is considered linearly dependent.

5. Can a set of two vectors be linearly dependent in three-dimensional space?

Yes, a set of two vectors can be linearly dependent in three-dimensional space. This means that the two vectors lie on the same plane and one can be written as a linear combination of the other. In general, a set of n vectors can be linearly dependent in n-dimensional space if one vector can be written as a linear combination of the others.

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