Why is the basis of a column space linearly independent 

The basis of a column space is linearly independent by definition, and here's why:

### Understanding the Basis of a Column Space
A **basis** for a vector space (like the column space of a matrix) is defined as a set of vectors that:
1. **Spans** the space: Every vector in the space can be expressed as a linear combination of the basis vectors.
2. Is **linearly independent**: No vector in the basis can be written as a linear combination of the other vectors in the set.

### Linear Independence Explained
A set of vectors is **linearly independent** if the only solution to the equation:
\[
c_1 \mathbf{v}_1 + c_2 \mathbf{v}_2 + \dots + c_n \mathbf{v}_n = \mathbf{0}
\]
is \( c_1 = c_2 = \dots = c_n = 0 \). In other words, the vectors in the set cannot be combined using non-zero coefficients to form the zero vector.

### Why Must the Basis Be Linearly Independent?
If a set of vectors spans a space but is not linearly independent, then at least one of the vectors can be written as a linear combination of the others. This would mean the set contains "redundant" vectors, making it unnecessarily large. A basis, by definition, should be the smallest possible set that still spans the space.

To formally define the concept:
- **Spanning** ensures that the entire space is covered by the set.
- **Linear independence** ensures that the set is as small as possible while still spanning the space.

### Example to Illustrate
Consider a set of vectors in \(\mathbb{R}^2\):
\[
\text{Set 1: } \left\{ \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \end{pmatrix} \right\}
\]
This set spans \(\mathbb{R}^2\) and is linearly independent, so it's a basis.

Now, consider:
\[
\text{Set 2: } \left\{ \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 2 \\ 0 \end{pmatrix} \right\}
\]
This set is **not** linearly independent because the second vector is a multiple of the first. Hence, it's not a basis since it doesn't meet the independence criterion.

### Conclusion
A basis must be linearly independent because it needs to efficiently and uniquely represent every vector in the space without any redundancy. If the vectors in the basis were dependent, they could be reduced to a smaller set that still spans the space, contradicting the definition of a basis.

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Do you have any other questions or need further clarification?

Here are some related questions to consider:
1. How do you check if a set of vectors is linearly independent?
2. Can the basis for a vector space be unique?
3. What happens if a set of vectors is linearly dependent?
4. How does linear independence relate to the rank of a matrix?
5. What is the significance of the dimension of a vector space?
6. How can you determine the number of vectors in a basis?
7. Can a linearly independent set be larger than the space it spans?
8. How does linear independence relate to eigenvectors?

**Tip:** Always test for linear independence by checking if the determinant of the matrix formed by the vectors is non-zero (for square matrices) or by solving the linear combination equation.

Explore why the basis of a column space is required to be linearly independent in linear algebra. This explanation covers the fundamental concepts of vector spaces, spanning, and linear independence, with practical examples. Ideal for college-level students studying linear algebra.

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