Linear transformation of 3 linearly Independence factors yields three linearly independent vectors. If false give counter

The statement is **true**. If you apply a **linear transformation** to three **linearly independent vectors** in a vector space, the resulting vectors will remain linearly independent, **provided the transformation is invertible (i.e., non-singular)**.

### Explanation:
Let’s assume you have three linearly independent vectors \( v_1, v_2, v_3 \) in a vector space \( V \), and you apply a linear transformation \( T: V \to W \) to these vectors. The transformation yields the vectors \( T(v_1), T(v_2), T(v_3) \).

For the vectors \( T(v_1), T(v_2), T(v_3) \) to remain linearly independent, the transformation \( T \) must be **invertible** (i.e., its matrix representation must have full rank, and its determinant should not be zero). If \( T \) is invertible, then it preserves the linear independence of the vectors.

#### Counterexample (Non-Invertible Case):
If \( T \) is **not invertible** (i.e., it has a rank deficiency or is singular), it could map linearly independent vectors to linearly dependent ones or even to the zero vector, thus violating the condition.

For example, consider the linear transformation \( T: \mathbb{R}^3 \to \mathbb{R}^2 \), represented by the matrix:

\[
T = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \end{pmatrix}
\]

Now apply this transformation to three linearly independent vectors in \( \mathbb{R}^3 \):

\[
v_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}, \quad
v_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}, \quad
v_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}
\]

The transformation gives:

\[
T(v_1) = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad
T(v_2) = \begin{pmatrix} 0 \\ 1 \end{pmatrix}, \quad
T(v_3) = \begin{pmatrix} 0 \\ 0 \end{pmatrix}
\]

Here, \( T(v_1) \) and \( T(v_2) \) are still linearly independent, but \( T(v_3) \) is the zero vector, which makes \( T(v_1), T(v_2), T(v_3) \) linearly dependent. This shows that if the transformation is not invertible, the resulting vectors may not remain linearly independent.

### In summary:
- If \( T \) is invertible, the resulting vectors will be linearly independent.
- If \( T \) is not invertible, the resulting vectors may be linearly dependent.

Would you like more details or clarification?

Here are 5 related questions to consider:
1. What is the condition for a matrix to be invertible?
2. How can you verify whether vectors are linearly independent?
3. How does the rank of a matrix affect the image of a transformation?
4. What happens to the dimension of a vector space after applying a linear transformation?
5. How do eigenvalues relate to the invertibility of a transformation?

**Tip:** When dealing with linear transformations, always check if the transformation is invertible to ensure properties like linear independence are preserved!

Linear transformation of 3 linearly independent factors yields three linearly independent vectors. If false, give counter.

Understand how applying a linear transformation to three linearly independent vectors yields three linearly independent vectors if the transformation is invertible. Includes a detailed explanation and a counterexample for non-invertible transformations. Suitable for undergraduate students studying linear algebra.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation