Linear independence

-

We will work with row vectors in \(\mathbb{R}^n\). Consider a set of \(m\) vectors. If one of these vectors can be expressed as a linear combination of the other vectors, these vectors are said to be linearly dependent. If none of these vectors can be expressed as a linear combination of the other vectors, they are said to be linearly independent. We show that the determinant of a matrix is zero if its row vectors are linearly dependent and vice versa.



Definition (Linear dependence)

We say that a finite sequence \(\mathbf{v}_1, \mathbf{v}_2, \cdots, \mathbf{v}_m\) of vectors is linearly dependent if there are real numbers \(\lambda_1, \lambda_2, \cdots, \lambda_m\) which are not all 0 (i.e., some are non-zero), such that

\[\sum_{i=1}^{m}\lambda_i\mathbf{v}_i = 0.\]

Why this terminology? Suppose \(\lambda_1 \neq 0\). Then we can rearrange the above equation into

\[\mathbf{v}_1 = -(1/\lambda_1)(\lambda_2\mathbf{v}_2 + \cdots + \lambda_m\mathbf{v}_m).\]

That is, \(\mathbf{v}_1\) can be expressed as a linear combination of the other vectors. In this sense, the vector \(\mathbf{v}_1\) depends on the other vectors.

Example. \((2, -1, 0), (-3, 2, 1),\) and \((7, -4, -1)\) are linearly dependent because

\[2(2, -1, 0) - 1\cdot(-3, 2, 1) -1\cdot(7, -4, -1) = (0, 0, 0).\] □

Definition (Linear independence)

If a finite sequence \(\mathbf{v}_1, \mathbf{v}_2, \cdots, \mathbf{v}_m\) of vectors is not linearly dependent, we say that it is linearly independent.

Remark. Therefore, \(\mathbf{v}_1, \mathbf{v}_2, \cdots, \mathbf{v}_m\) are linearly independent if and only if

\[\sum_{i=1}^{m}\lambda_i\mathbf{v}_i = 0\]

implies \[\lambda_1 = \lambda_2 = \cdots = \lambda_m = 0.\] □

Example. \(\mathbf{e}_1 = (1, 0)\) and \(\mathbf{e}_2 = (0, 1)\) are linearly independent vectors. (Verify this!) □

Theorem

Suppose that \(\mathbf{v}_1, \mathbf{v}_2, \cdots, \mathbf{v}_m\) are linearly independent. If there is a vector \(\mathbf{u}\) such that

\[\mathbf{u} = \sum_{i=1}^{m}\beta_i \mathbf{v}_i\]

for some \(\beta_1, \beta_2, \cdots, \beta_m\), then this representation is unique.

Remark. What this uniqueness means is the following. If \(\mathbf{u}\) can be represented in terms of another linear combination of \(\mathbf{v}_i\), say,

\[\mathbf{u} = \sum_{i=1}^{m}\gamma_i \mathbf{v}_i,\]

then, we have \(\beta_1 = \gamma_1, \beta_2 = \gamma_2, \cdots, \beta_m = \gamma_m.\) □

Proof. Suppose there is an alternative representation

\[\mathbf{u} = \sum_{i=1}^{m}\gamma_i \mathbf{v}_i.\]

Then subtracting both sides, we have

\[\mathbf{0} = \mathbf{u} -\mathbf{u} = \sum_{i=1}^{m}(\beta_i - \gamma_i)\mathbf{v}_i.\]

Since \(\mathbf{v}_1, \mathbf{v}_2, \cdots, \mathbf{v}_m\) are linearly independent by assumption, it follows that \(\beta_i - \gamma_i = 0\), and hence \(\beta_i = \gamma_i\) for all \(i = 1, 2, \cdots, m\) as required. ■

Linear independence and matrix determinant

Theorem

Let \[ A = \begin{pmatrix} \mathbf{a}_1\\ \mathbf{a}_2\\ \vdots\\ \mathbf{a}_n \end{pmatrix} \]

be an \(n\times n\) matrix where \(\mathbf{a}_1, \mathbf{a}_2,\cdots, \mathbf{a}_n\) are \(n\)-dimensional row vectors. If these row vectors are linearly dependent, then \(\det A = 0\).

Proof. Since the row vectors are linearly dependent, at least one of them can be expressed as a linear combination of the other vectors. Without the loss of generality, assume \(\mathbf{a}_n = \sum_{i=1}^{n-1}\lambda_i\mathbf{a}_i\) with not all \(\lambda_1, \cdots, \lambda_{n-1}\) being equal to 0.

Then, by the properties of determinants, we have \[ \begin{eqnarray} \begin{vmatrix} \mathbf{a}_1\\ \mathbf{a}_2\\ \vdots\\ \mathbf{a}_{n-1}\\ \mathbf{a}_n \end{vmatrix} &=& \begin{vmatrix} \mathbf{a}_1\\ \mathbf{a}_2\\ \vdots\\ \mathbf{a}_{n-1}\\ \lambda_1\mathbf{a}_1 + \cdots +\lambda_{n-1}\mathbf{a}_{n-1} \end{vmatrix}\\ &=& \sum_{i=1}^{n-1}% \begin{vmatrix} \mathbf{a}_1\\ \mathbf{a}_2\\ \vdots\\ \mathbf{a}_{n-1}\\ \lambda_i\mathbf{a}_i \end{vmatrix}\\ &=& \sum_{i=1}^{n-1}% \lambda_i\begin{vmatrix} \mathbf{a}_1\\ \mathbf{a}_2\\ \vdots\\ \mathbf{a}_{n-1}\\ \mathbf{a}_i \end{vmatrix}. \end{eqnarray} \]

In the last sum, each determinant contains two rows of the same vectors (i.e., \(\mathbf{a}_i\) is equal to one of \(\mathbf{a}_1,\cdots, \mathbf{a}_{n-1}\)). Therefore, by the property of determinants, all the terms are 0. ■

See also: Review More on determinants for the properties of determinants.

Example. Consider the matrix determinant \[ \begin{vmatrix} 2 & -1 & 0\\ -3 & 2 & 1\\ 7 & -4 & -1 \end{vmatrix}. \] We know that the row vectors are linearly dependent (see the example above), and hence, the determinant is zero. □

Corollary 

Let

\[A = \begin{pmatrix} \mathbf{a}_1\\ \mathbf{a}_2\\ \vdots\\ \mathbf{a}_n \end{pmatrix} \]

be an \(n\times n\) matrix where \(\mathbf{a}_1, \mathbf{a}_2,\cdots, \mathbf{a}_n\) are \(n\)-dimensional row vectors. If \(\det A \neq 0\), then these row vectors are linearly independent.

Proof. The contrapositive of the above theorem. ■

The converse is also true: If the row vectors are linearly independent, then \(\det A \neq 0\). However, proving this is beyond the scope of this module. See some textbooks on Linear Algebra (We need to define the notion of the rank of a matrix).


Comments

Post a Comment

Popular posts from this blog

Open sets and closed sets in \(\mathbb{R}^n\)

-
Image
Open sets In \(\mathbb{R}\), we have the notion of an open interval such as \((a, b) = \{x \in \mathbb{R} | a < x < b\}\). We want to extend this idea to apply to \(\mathbb{R}^n\). We also introduce the notions of bounded sets and closed sets in \(\mathbb{R}^n\). Recall that the \(\varepsilon\)-neighbor of a point \(x\in\mathbb{R}^n\) is defined as \(N_{\varepsilon}(x) = \{y \in \mathbb{R}^n | d(x, y) < \varepsilon \}\) where \(d(x,y)\) is the distance between \(x\) and \(y\). Definition (Open set) A subset \(U\) of \(\mathbb{R}^n\) is said to be an  open set  if the following holds: \[\forall x \in U ~ \exists \delta > 0 ~ (N_{\delta}(x) \subset U).\tag{Eq:OpenSet}\] That is, for every point in an open set \(U\), we can always find an open ball centered at that point, that is included in \(U\). See the following figure. Perhaps, it is instructive to see what is  not  an open set. Negating (Eq:OpenSet), we have \[\exists x \in U ~ \forall \delta > 0 ~ (N_{\delta}(x) \not
Read more

Euclidean spaces

-
Image
We would like to study multivariate functions (i.e., functions of many variables), continuous multivariate functions in particular. To define continuity, we need a measure of "closeness" between points. One measure of closeness is the Euclidean distance. The set \(\mathbb{R}^n\) (with \(n \in \mathbb{N}\)) with the Euclidean distance function is called a Euclidean space. This is the space where our functions of interest live. The real line is a geometric representation of \(\mathbb{R}\), the set of all real numbers. That is, each \(a \in \mathbb{R}\) is represented as the point \(a\) on the real line. The coordinate plane , or the \(x\)-\(y\) plane , is a geometric representation of \(\mathbb{R}^2\), the set of all pairs of real numbers. Each pair of real numbers \((a, b)\) is visualized as the point \((a, b)\) in the plane. Remark . Recall that \(\mathbb{R}^2 = \mathbb{R}\times\mathbb{R} = \{(x, y) | x, y \in \mathbb{R}\}\) is the Cartesian product of \(\mathbb{R}\) with i
Read more

Applications of multiple integrals

-
Image
We can use multiple integrals to compute areas and volumes of various shapes. Area of a planar region Definition (Area) Let \(D\) be a bounded closed region in \(\mathbb{R}^2\). \(D\) is said to have an area if the multiple integral of the constant function 1 over \(D\), \(\iint_Ddxdy\), exists. Its value is denoted by \(\mu(D)\): \[\mu(D) = \iint_Ddxdy.\] Example . Let us calculate the area of the disk \(D = \{(x,y)\mid x^2 + y^2 \leq a^2\}\). Using the polar coordinates, \(x = r\cos\theta, y = r\sin\theta\), \(dxdy = rdrd\theta\), and the ranges of \(r\) and \(\theta\) are \([0, a]\) and \([0, 2\pi]\), respectively. Thus, \[\begin{eqnarray*} \mu(D) &=& \iint_Ddxdy\\ &=&\int_0^a\left(\int_0^{2\pi}rd\theta\right)dr\\ &=&2\pi\int_0^a rdr\\ &=&2\pi\left[\frac{r^2}{2}\right]_0^a = \pi a^2. \end{eqnarray*}\] □ Volume of a solid figure Definition (Volume) Let \(V\) be a solid figure in the \((x,y,z)\) space \(\mathbb{R}^3\). \(V\) is sai
Read more