Matrix Algebra

Linear Independence, Basis

Linear independence. Let $\mathbf{v}_1, \ldots, \mathbf{v}_k$ be column vectors in $\mathbb{R}^n$ . Then we have the homogeneous equation

$\displaystyle x_1 \mathbf{v}_{1} + \cdots + x_k \mathbf{v}_{k} = \mathbf{0}$

in the form of vector equation. If the above homogeneous equation has only the trivial solution

$\displaystyle x_1 = \cdots = x_k = 0$

then the vectors $\mathbf{v}_1, \ldots, \mathbf{v}_k$ are said to be linearly independent; otherwise, they are linearly dependent.

EXAMPLES 2. Determine whether the vectors $\mathbf{v}_1$ , $\mathbf{v}_2$ and $\mathbf{v}_3$ are linearly independent or not in each of the following.

$\mathbf{v}_1 = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$ , $\mathbf{v}_2 = \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix}$ , and $\mathbf{v}_3 = \begin{bmatrix} 2 \\ 1 \\ 0 \end{bmatrix}$
$\mathbf{v}_1 = \begin{bmatrix} 0 \\ 1 \\ 5 \end{bmatrix}$ , $\mathbf{v}_2 = \begin{bmatrix} 1 \\ 2 \\ 8 \end{bmatrix}$ , and $\mathbf{v}_3 = \begin{bmatrix} 4 \\ -1 \\ 0 \end{bmatrix}$

Basis. Suppose that a subspace

$\displaystyle V = \mathrm{span}\{\mathbf{v}_1, \ldots, \mathbf{v}_k\}$

is spanned by $\mathbf{v}_1, \ldots, \mathbf{v}_k$ , and that $\mathbf{v}_1, \ldots, \mathbf{v}_k$ are linearly independent. Then the collection

$\displaystyle \mathcal{B} = \{\mathbf{v}_1, \ldots, \mathbf{v}_k\}$

of vectors is called a basis for the subspace

, and the subspace

is said to be k-dimensional. In short we write

$\displaystyle \dim V = k.$

As a special case we posit $\dim\{\mathbf{0}\} = 0$ . That is, the zero subspace $\{\mathbf{0}\}$ is 0-dimensional.

Representation theorem. Let $\mathbf{v}_1, \ldots, \mathbf{v}_k$ be a basis for the vector space . Then for each $\mathbf{x}\in V$ (that is, for each $\mathbf{x}$ in ) there exists a unique representation

$\displaystyle \mathbf{x} = c_1 \mathbf{v}_{1} + \cdots + c_k \mathbf{v}_{k}$

with scalars $c_1,\ldots,c_k$ .

EXAMPLES 7. Let be the subspace spanned by $\mathbf{v}_1 = \begin{bmatrix} 3 \\ 6 \\ 2 \end{bmatrix}$ and $\mathbf{v}_2 = \begin{bmatrix} -1 \\ 0 \\ 1 \end{bmatrix}$ . Determine whether $\mathbf{x} = \begin{bmatrix} 3 \\ 12 \\ 7 \end{bmatrix}$ is in . If so, find the unique representation by $\mathbf{v}_1$ and $\mathbf{v}_2$ .