Vectors and Matrices

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In their elementary form, matrices and vectors are just lists of real numbers in different formats. An $n$ -vector is a list of $n$ numbers $(x_1,x_2,\ldots ,x_n)$ . We may write this vector as a row vector as we have just done — or as a column vector $\vect {x}{n}.$ The set of all (real-valued) $n$ -vectors is denoted by $\R ^n$ ; so points in $\R ^n$ are called vectors. The sets $\R ^n$ when $n$ is small are very familiar sets. The set $\R ^1=\R$ is the real number line, and the set $\R ^2$ is the Cartesian plane. The set $\R ^3$ consists of points or vectors in three dimensional space.

An $m\times n$ matrix is a rectangular array of numbers with $m$ rows and $n$ columns. A general $2\times 3$ matrix has the form $A=\left (\begin {array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end {array}\right ).$ We use the convention that matrix entries $a_{ij}$ are indexed so that the first subscript $i$ refers to the row while the second subscript $j$ refers to the column. So the entry $a_{21}$ refers to the matrix entry in the $2^{nd}$ row, $1^{st}$ column.

An $n\times m$ matrix $A$ and an $n'\times m'$ matrix $B$ are equal precisely when the sizes of the matrices are equal ( $n=n'$ and $m=m'$ ) and when each of the corresponding entries are equal ( $a_{ij}=b_{ij}$ ).

There is some redundancy in the use of the terms “vector” and “matrix”. For example, a row $n$ -vector may be thought of as a $1\times n$ matrix, and a column $n$ -vector may be thought of as a $n\times 1$ matrix. There are situations where matrix notation is preferable to vector notation and vice-versa.

Addition and Scalar Multiplication of Vectors

There are two basic operations on vectors: addition and scalar multiplication. Let $x=(x_1,\ldots ,x_n)$ and $y=(y_1,\ldots ,y_n)$ be $n$ -vectors. Then $x+y=(x_1+y_1,\ldots ,x_n+y_n);$ that is, vector addition is defined as componentwise addition.

Similarly, scalar multiplication is defined as componentwise multiplication. A scalar is just a number. Initially, we use the term scalar to refer to a real number — but later on we sometimes use the term scalar to refer to a complex number. Suppose $r$ is a real number; then the multiplication of a vector by the scalar $r$ is defined as $rx = (rx_1,\ldots ,rx_n).$

Subtraction of vectors is defined simply as $x-y = (x_1-y_1,\ldots ,x_n-y_n).$ Formally, subtraction of vectors may also be defined as $x-y = x+(-1)y.$ Division of a vector $x$ by a scalar $r$ is defined to be $\frac {1}{r} x.$ The standard difficulties concerning division by zero still hold.

Addition and Scalar Multiplication of Matrices

Similarly, we add two $m\times n$ matrices by adding corresponding entries, and we multiply a scalar times a matrix by multiplying each entry of the matrix by that scalar. For example, $\mattwo {0}{2}{4}{6} + \mattwo {1}{-3}{1}{4} = \mattwo {1}{-1}{5}{10}$ and $4\mattwo {2}{-4}{3}{1} = \mattwo {8}{-16}{12}{4}.$ The main restriction on adding two matrices is that the matrices must be of the same size. So you cannot add a $4\times 3$ matrix to $6\times 2$ matrix — even though they both have twelve entries.

Exercises

In Exercises ?? – ??, let $x=(2,1,3)$ and $y=(1,1,-1)$ and compute the given expression.

$x+y\begin {prompt} = \left (\answer {3},\answer {2},\answer {2}\right ). \end {prompt}$ .

$2x-3y\begin {prompt} = \left (\answer {1},\answer {-1},\answer {9}\right ) \end {prompt}$ .

$2x - 3y = (4,2,6) - (3,3,-3)$ .

$(4,2,6) - (3,3,-3) = (1,-1,9)$ .

$4x\begin {prompt} = \left (\answer {8},\answer {4},\answer {12}\right ) \end {prompt}$ .

Let $A$ be the $3\times 4$ matrix $A=\left (\begin {array}{rrrr} 2 & -1 & 0 & 1 \\ 3 & 4 & -7 & 10\\ 6 & -3 & 4 & 2 \end {array}\right ).$

(a): For which $n$ is a row of $A$ a vector in $\R ^n$ ? $n = \answer {4}$ .
(b): What is the $2^{nd}$ column of $A$ ? $\left (\begin {array}{r} \answer {-1} \\ \answer {4} \\ \answer {-3} \end {array} \right )$
(c): Let $a_{ij}$ be the entry of $A$ in the $i^{th}$ row and the $j^{th}$ column. What is $a_{23}-a_{31}$ ? $a_{23}-a_{31} = \answer {-13}.$

For each of the pairs of vectors or matrices in Exercises ?? – ??, decide whether addition of the members of the pair is possible; and, if addition is possible, perform the addition.

$x=(2,1)$ and $y=(3,-1)$ .