Linear Independence and Basis in R^3

Understanding Span

The span of a set of vectors is the collection of all possible linear combinations that can be created from them. If you imagine two vectors in $\mathbb{R}^3$ that are not pointing in the same direction, their span is typically a 2D plane passing through the origin. If you have three vectors, their span might be the entire 3D space ($\mathbb{R}^3$).

To determine if a specific vector is in the span of a set, we attempt to solve the equation:

$$C_1y_1 + C_2y_2 + \ldots + C_ny_n = \text{target vector}$$

If we can find scalars $C_i$ that make this equation true, the target vector is in the span. If the system of equations has a solution for any arbitrary $(x, y, z)$, then the set of vectors spans $\mathbb{R}^3$.

Linear Independence vs. Dependence

Not all sets of vectors are created equal. Some vectors provide "new" directions, while others are redundant. This is the distinction between linearly independent and linearly dependent sets.

A set of vectors $\{y_1, y_2, \ldots, y_n\}$ is linearly independent if the only way to satisfy the equation:

$$C_1y_1 + C_2y_2 + \ldots + C_ny_n = 0$$

is by setting all scalars to zero ($C_1 = C_2 = \ldots = C_n = 0$). This is known as the trivial solution. If there is any other solution where at least one scalar is non-zero, the vectors are linearly dependent.

Exam Tip: If you have more vectors than there are dimensions (for example, 4 vectors in $\mathbb{R}^3$), the set is automatically linearly dependent because there will always be more unknowns than equations, leading to infinitely many solutions.

The Determinant Shortcut

When dealing with a square set of vectors (e.g., three vectors in $\mathbb{R}^3$), we can use the determinant of the coefficient matrix to quickly check both span and independence.

1. If $\det(A) \neq 0$, the vectors are linearly independent and they span the space.
2. If $\det(A) = 0$, the vectors are linearly dependent and they do not span the space.

Worked Example: Using the Determinant

Consider the vectors $y_1=(2,0,1), y_2=(-1,3,4), y_3=(1,1,-2)$. To see if they span $\mathbb{R}^3$, we set up the matrix:

$$A = \begin{bmatrix} 2 & -1 & 1 \\ 0 & 3 & 1 \\ 1 & 4 & -2 \end{bmatrix}$$

Calculating the determinant:

$$\det A = 2((3)(-2) - (4)(1)) - (-1)((0)(-2) - (1)(1)) + 1((0)(4) - (1)(3))$$ $$\det A = 2(-6 - 4) + 1(0 - 1) + 1(0 - 3) = -20 - 1 - 3 = -24$$

Since $\det A \neq 0$, these vectors are linearly independent and span $\mathbb{R}^3$.

Defining a Basis

A basis is the "gold standard" for describing a vector space. A set of vectors $S$ is a basis for a vector space $V$ if it satisfies two specific conditions:

1. The vectors in $S$ are linearly independent.
2. The vectors in $S$ span $V$.

In $\mathbb{R}^3$, any basis must contain exactly three vectors. If you have only two vectors, they might be independent, but they can only span a plane, not the whole of $\mathbb{R}^3$. If you have four vectors, they might span $\mathbb{R}^3$, but they cannot be a basis because they will be linearly dependent.

The number of vectors in a basis is called the dimension of the vector space. For $\mathbb{R}^3$, the dimension is 3. The most common example is the standard unit vectors:

$$e_1 = (1, 0, 0), \quad e_2 = (0, 1, 0), \quad e_3 = (0, 0, 1)$$

These vectors are linearly independent and span $\mathbb{R}^3$, making them a basis.