Contents

2.1 Vectors

Operations

Vector Addition

The addition of vectors occurs component-wise. For two vectors in \(\mathbb R^4\):

\[\begin{split}\vec v = \left[\begin{array}{r}3\\-1\\2\\9\\\end{array}\right], \vec u = \left[\begin{array}{r}4\\5\\-2\\-5\\\end{array}\right], \hspace{1cm}\vec v + \vec u = \left[\begin{array}{r}7\\4\\0\\4\\\end{array}\right]\end{split}\]

If two vectors have a different number of components, they cannot be added or subtracted.

Scalar Multiplication

Suppose we take a real number \(r\). We can multiply a vector by this scalar by multiplying each component of the vector by \(r\). For example, if \(r = 3\), then:

\[\begin{split}r\vec v = 3\left[\begin{array}{r}3\\-1\\2\\9\\\end{array}\right] = \left[\begin{array}{r}9\\-3\\6\\27\\\end{array}\right]\end{split}\]

Vector Subtraction

To substract \(\vec v - \vec u\) we add \(\vec v\) to \((-1)\vec u\).

\[\begin{split}\vec v - \vec u = \left[\begin{array}{r}3\\-1\\2\\9\\\end{array}\right] + \left[\begin{array}{r}-4\\-5\\2\\5\\\end{array}\right] = \left[\begin{array}{r}-1\\-6\\4\\14\\\end{array}\right]\end{split}\]

Clearly, like addition, we are subtracting component-wise.

MATLAB and Vectors

Let’s create the vectors used in the above examples.

v = [3 ; -1 ; 2 ; 9]
u = [4 ; 5 ; -2 ; -5]

v =

     3
    -1
     2
     9


u =

     4
     5
    -2
    -5

Addition and subtraction work exactly as you would guess.

v + u

ans =

     7
     4
     0
     4

v - u

ans =

    -1
    -6
     4
    14

Scalar multiplication also makes sense, for example, if \(r = 3\).

3 * v

ans =

     9
    -3
     6
    27

Be sure to spend time reading and interacting Margalit and Robanoff textbook to visualize what it means to add, subtract vectors and to take the scalar multiple of a vector.

Note

In this course, the word scalar amost always means real number. However, we can have complex-valued matrices and vectors. In those cases, a scalar would be a complex number.

Linear Combinations of Vectors

A linear combination of the vectors \(\vec v_1, \vec v_2, \dots \vec v_n\) uses scalars \(c_1, c_2, \dots c_n\) as weights. If we have the vectors

\[\begin{split}\vec v_1 = \left[\begin{array}{r}5\\3\\-2\\\end{array}\right], \vec v_2 = \left[\begin{array}{r}0\\2\\5\\\end{array}\right], \vec v_3 = \left[\begin{array}{r}3\\4\\3\\\end{array}\right]\end{split}\]

and weights \(c_1 = 4, c_2 = -1, c_3 = 2\), then the result is the linear combination:

\[\begin{split}c_1\vec v_1 + c_2\vec v_2 +c_3\vec v_3 = 4\left[\begin{array}{r}5\\3\\-2\\\end{array}\right]-\left[\begin{array}{r}0\\2\\5\\\end{array}\right] +2\left[\begin{array}{r}3\\4\\3\\\end{array}\right] = \left[\begin{array}{r}26\\18\\-7\\\end{array}\right]\end{split}\]

Linear Combinations in MATLAB

Let’s create the vectors from the above example.

v1 = [5 ; 3 ; -2]
v2 = [ 0; 2 ; 5 ]
v3 = [3 ; 4 ; 3]

v1 =

     5
     3
    -2


v2 =

     0
     2
     5


v3 =

     3
     4
     3

Then we apply addition, subtraction and scalar multiplications as before.

4 * v1 - v2 + 2 * v3

ans =

    26
    18
    -7
Chapter 2 2.2 Vector Equations and Spans