# Proof involving linear transformation of a set of vectors

• Lanthanum
In summary, if S is linearly dependent, then the set {T(u), T(v), T(w)} is also linearly dependent under the linear transformation T. This is because there exists a linear relationship between the vectors in S, and this relationship holds true even after applying the transformation T. Therefore, S is also linearly dependent under T.
Lanthanum

## Homework Statement

Let T:$$\Re^{n}$$$$\rightarrow$$$$\Re^{m}$$ and let S={u,v,w}$$\in$$$$\Re^{n}$$.

If S is linearly dependent, show that {T(u), T(v), T(w)} is also linearly dependent.

N/A

## The Attempt at a Solution

Since S$$\in$$$$\Re^{n}$$ then S$$\in$$$$\Re^{m}$$.
Not sure where to go from here

If the set of vectors are dependent you can find a linear relationship between them.

What happens to this linear relationship under the map T?

If p(t) is the linear relationship between the vectors in S, does that mean that T[p(t)] is the linear relationship between the vectors in S?

If so I guess that proves that S is also linearly dependent since there exists the linear relationship T[p(t)] between its elements.

Look up the definition of linear dependence.

Something is lacking in your statement of the problem. Is T supposed to be a linear map? You did not say so.

Lanthanum said:
If p(t) is the linear relationship between the vectors in S, does that mean that T[p(t)] is the linear relationship between the vectors in S?

If so I guess that proves that S is also linearly dependent since there exists the linear relationship T[p(t)] between its elements.
Don't ask, work it out. Your "p(t) is the linear relationship" is what Ziox suggested you look up- the definition of "linearly dependent". If vectors, u, v, and w are linearly dependent, then there exist numbers, a, b, c, not all 0, such that au+ bv+ cw= 0.

Now look at T(au+ bv+ cw)= T(0).

As Arkajad suggested, you will need to specify that T is a linear transformation (which you do in a later post).

Aha! I think I have it.

If vectors, u, v, and w are linearly dependent, then there exist numbers, a, b, c, not all 0, such that au+ bv+ cw= 0.

Now look at T(au+ bv+ cw)= T(0)

Since T is a linear transformation, T(0)=0, and T(au+ bv+ cw)=aT(u)+bT(v)+cT(w)=0

Since there exists scalars, a, b, c, not all 0, such that the above equation is true, S` is linearly dependent.

I can't believe I missed that even after ZioX's advice, guess I needed a study break. Thanks guys.

## 1. What is a linear transformation?

A linear transformation is a function that maps one vector space to another in a way that preserves vector addition and scalar multiplication. In other words, the transformation follows the properties of linearity, such as preserving parallelism and maintaining the origin.

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

To prove that a transformation is linear, you need to 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 scalar multiple of a vector is equal to the scalar multiple of the transformation of the original vector.

## 3. Can a linear transformation change the dimension of a vector space?

No, a linear transformation cannot change the dimension of a vector space. This is because it must preserve the properties of linearity, including the number of dimensions in the vector space. So, if the original vector space has n dimensions, the transformed vector space must also have n dimensions.

## 4. How do you prove that a set of vectors is linearly independent?

A set of vectors is linearly independent if none of the vectors can be expressed as a linear combination of the other vectors in the set. To prove this, you can use the definition of linear independence and set up a system of equations to show that the only solution is the trivial solution (all coefficients equal to 0).

## 5. Is there a relationship between linear transformations and matrices?

Yes, there is a close relationship between linear transformations and matrices. Every linear transformation can be represented by a matrix, and every matrix can be thought of as a linear transformation. This makes it easier to perform calculations involving linear transformations by using matrix operations.

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