Row vectors, column vectors, and matrices

-

We define an (\(n\)-dimensional) (row) vector to be an element of \(\mathbb{R}^n\) endowed with addition and scalar multiplication. 



So, a row vector is something like \((x_1, x_2, \cdots, x_n)\). We may omit the commas as in \((x_1~ x_2 ~ \cdots ~ x_n)\). But vectors are not just elements of \(\mathbb{R}^n\). We define addition between row vectors by

\[(x_1, \cdots, x_n) + (y_1, \cdots, y_n) = (x_1+y_1, \cdots, x_n + y_n).\tag{eq:vadd}\]

We also define multiplication by a real number \(\lambda\in\mathbb{R}\) by

\[\lambda\cdot(x_1,\cdots, x_n) = (\lambda x_1, \cdots, \lambda x_n).\tag{eq:svmul}\]

In this context, the real numbers are called scalars because it scales the vector.

In summary, vectors are elements of \(\mathbb{R}^n\) where addition and scalar multiplications are defined. We will give more characterization of vectors in later posts.

Example. \((1, 2), (-5, 3) \in \mathbb{R}^2\) are 2-dimensional row vectors. We can add them as

\[(1, 2) + (-5, 3) = (1-5, 2+3) = (-4, 5).\]

We can multiply them by scalars (real numbers):

\[\begin{eqnarray*} \pi\cdot(1, 2) &=& (\pi, 2\pi),\\ -1\cdot(-5, 3) &=& (5, -3). \end{eqnarray*}\]

□

Instead of writing \(n\) real numbers in a row like \((x_1, \cdots, x_n)\), we could write them in a column like

\[\begin{pmatrix}x_1\\\vdots\\x_n\end{pmatrix}\]

which we call a column vector. Addition and scalar multiplication can be defined similarly to row vectors.


If \(n\) is not prime, we can write a point in \(\mathbb{R}^n\) in multiple rows and columns. For example, a point in \(\mathbb{R}^6\) may be written as

\[(x_1, x_2, x_3, x_4, x_5, x_6)\]

or

\[\begin{pmatrix}x_1\\ x_2\\ x_3\\x_4\\ x_5\\ x_6\end{pmatrix}\]

or

\[\begin{pmatrix}x_1 & x_2 & x_3\\x_4 & x_5 & x_6\end{pmatrix}\]

or

\[\begin{pmatrix}x_1 & x_2\\x_3 & x_4\\x_5 & x_6\end{pmatrix}.\]

Such a rectangular array of numbers is called a matrix. If there are \(m\) rows and \(n\) columns in a matrix, we call it an \(m\times n\) matrix and regard it as a member of the set \(\mathbb{R}^{m\times n}\). We can also define element-wise addition and scalar multiplication for matrices. Now we can see that an \(n\)-dimensional row vector is really a \(1\times n\) matrix, and an \(n\)-dimensional column vector is an \(n \times 1\) matrix. Accordingly, we can define addition and scalar multiplication of matrices in the same way as vectors.

Example. 

\[\begin{pmatrix} 1 & 2 & 3\\ 4 & 5 & 6 \end{pmatrix} + \begin{pmatrix} -3 & -1 & 2\\ 1 & -2 & 3 \end{pmatrix} = \begin{pmatrix} -2 & 1 & 5\\ 5 & 3 & 9 \end{pmatrix},\]

\[5\cdot\begin{pmatrix} 1 & 2 & 3\\ 4 & 5 & 6 \end{pmatrix} = \begin{pmatrix} 5 & 10 & 15\\ 20 & 25 & 30 \end{pmatrix}.\]

Note that two matrices can be added if and only if they have the same shape. □

Remark. So far, we have worked with \(\mathbb{R}^n\) to define vectors (including matrices). We can similarly define vectors and matrices on \(\mathbb{C}^n\) or \(\mathbb{Q}^n\) where scalars are complex or rational numbers, respectively. Note that \(\mathbb{R}\), \(\mathbb{C}\) and \(\mathbb{Q}\) are fields. In fact, we may define vectors and matrices over any field. □

Definition (Linear combination)

Let \(\mathbf{v}_1, \mathbf{v}_2, \cdots, \mathbf{v}_n\) be vectors and \(\lambda_1, \lambda_2, \cdots, \lambda_n\) be scalars. The expression of the form

\[\lambda_1\mathbf{v}_1 + \lambda_2\mathbf{v}_2 + \cdots + \lambda_n\mathbf{v}_n\]

is called the linear combination of the vectors \(\mathbf{v}_1, \mathbf{v}_2, \cdots, \mathbf{v}_n\) with the coefficients \(\lambda_1, \lambda_2, \cdots, \lambda_n\).


Example. 

\[(5, -9) = 3(1, -1) + 2(1, -3).\]

Thus, \((5, -9)\) can be represented as a linear combination of \((1,-1)\) and \((1, -3)\) with coefficients 3 and 2, respectively. □


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