Does anyone know how to prove the following:

The set of images of a linearly independent set of vectors under a linear transformation are also linearly independent?

Thanks.

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- July 7th 2012, 01:41 PMPiIsCoolLinear Independence and Transformation Proof Help
Does anyone know how to prove the following:

The set of images of a linearly independent set of vectors under a linear transformation are also linearly independent?

Thanks. - July 7th 2012, 01:58 PMemakarovRe: Linear Independence and Transformation Proof Help
This is true only for injective transformations. For example, the projection of two-dimensional vectors on the horizontal axis maps any two vectors into linearly dependent vectors.

- July 7th 2012, 02:06 PMPiIsCoolRe: Linear Independence and Transformation Proof Help
What would be the best way to prove this?

- July 7th 2012, 02:10 PMemakarovRe: Linear Independence and Transformation Proof Help
Prove what? The original claim is false in general.

- July 7th 2012, 05:14 PMPiIsCoolRe: Linear Independence and Transformation Proof Help
What is the best way to answer part 2?

Prove that the set of images of a linearly dependent set of vectors under a linear transformation is linearly dependent. Let {v1¬, , … , vp} be a set of linearly dependent vectors, and let T: Rn Rm be a linear transformation. Show that {T(v1), …, T(vp)} is linearly dependent. Part 2: Is it also true that the images of a linearly independent set of vectors under a linear transformation is also linearly independent? Explain.

Thanks. - July 8th 2012, 04:37 AMemakarovRe: Linear Independence and Transformation Proof Help
- July 8th 2012, 08:18 AMDevenoRe: Linear Independence and Transformation Proof Help
suppose there exist c

_{1},c_{2},...,c_{n}not all 0 with:

c_{1}v_{1}+ c_{2}v_{2}+...+ c_{n}v_{n}= 0

(that is {v_{1},v_{2},...,v_{n}} is a linearly dependent set), and that T is a linear transformation.

then T(c_{1}v_{1}+ c_{2}v_{2}+...+ c_{n}v_{n}) = T(0) = 0.

but since T is linear:

0 = T(c_{1}v_{1}+ c_{2}v_{2}+...+ c_{n}v_{n}) = c_{1}T(v_{1}) + c_{2}T(v_{2}) +...+ c_{n}T(v_{n})

which shows that {T(v_{1}),T(v_{2}),...,T(v_{n})} is linearly dependent.

it is NOT true that the image of a linearly independent set under a linear transformation is linearly independent.

for example, if T is the 0-map, T(v) = 0, for all v in V, then even if B = {v_{1},v_{2},...,v_{n}} is a basis,

T(B) = {T(v_{1}),T(v_{2}),...,T(v_{n})} = {0,0,...,0} = {0} (since repeated elements of a SET don't "count extra"),

and {0} is ALWAYS a linearly dependent set.

some equivalent (sufficient) conditions for T(S) to be LI, when S is LI:

a) T is injective

b) det(T) ≠ 0

c) ker(T) = {0}

d) rank(T) = dim(V) (where V is the domain of T)

note these conditions are not necessary, it may well be that T(S) is LI when S is, even if none (thus all) of the above hold. for example, let T be multiplication by the matrix:

[1 0 0]

[0 1 0]

[0 0 0].

then for S = {(1,0,0),(0,1,0)}, T(S) is LI, even though T is singular. however, S' = {(0,1,0),(0,2,2)} is also LI, but T(S) = {(0,1,0),(0,2,0)}, which is NOT linearly independent, since the second vector is a scalar multiple of the first:

we have 2(0,1,0) + (-1)(0,2,0) = (0,0,0) and neither of {2,-1} is 0.