Linear independence is a fundamental concept in linear algebra that deals with vectors. A set of vectors is said to be linearly independent if no vector in the set can be expressed as a linear combination of the others.

Explanation:

• Linear Combination: A linear combination of vectors involves taking some scalar multiples of the vectors and adding them together. For example, if we have vectors $\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n$, then a linear combination would be something like $c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \ldots + c_n\mathbf{v}_n$, where $c_1, c_2, \ldots, c_n$ are scalars (real numbers, for example).

• Linearly Independent: The vectors $\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n$ are linearly independent if the only way to get the zero vector $\mathbf{0}$ as a linear combination of these vectors is by having all the scalars equal to zero. Mathematically, this means: $c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \ldots + c_n\mathbf{v}_n = \mathbf{0}$ implies that $c_1 = c_2 = \ldots = c_n = 0$.

• Linearly Dependent: If there exists a non-trivial solution (i.e., not all scalars are zero) to the equation above, then the vectors are said to be linearly dependent. This means at least one of the vectors can be written as a combination of the others.

Example:

Consider the vectors $\mathbf{v}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\mathbf{v}_2 = \begin{pmatrix} 2 \\ 0 \end{pmatrix}$. These vectors are linearly dependent because $\mathbf{v}_2$ can be written as $2\mathbf{v}_1$. However, the vectors $\mathbf{v}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $\mathbf{v}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ are linearly independent because no scalar multiple of one can give the other.

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Related Questions:

1. How do you determine if a set of vectors is linearly independent?
2. What is the significance of linear independence in vector spaces?
3. How does linear independence relate to the rank of a matrix?
4. What is the geometric interpretation of linear independence?
5. How does linear independence apply in solving systems of linear equations?

Tip: To test for linear independence, one effective method is to arrange the vectors as columns in a matrix and perform row reduction. If the matrix has a pivot in every column, the vectors are linearly independent.