1. Vectors

A matrix with only one column is called a column vector, or simply a vector.Examples of vectors with two entries are

u =

31

�, v =

2

4327

3

5 , w =

2

664

w1

w2

w3

w4

3

775 ,

where w1, w2, w3, w4 are any real numbers. The set of all vectors with two entries isdenoted by R2 (read “r-two”). The vector u above would be an example of this. Avector with n entries is a vector in Rn.

The R stands for the real numbers that appear as entries in the vectors, and theexponent 2 indicates that each vector contains two entries. Two vectors in R2 (andRn) are equal if and only if their corresponding entries are equal.

If two vectors are the same size, we can add them (or subtract) component-wise:31

�+

23

�=

�.

We can also scale vectors by multiplying vectors by a scalar number:

5

23

�=

�.

Important: we cannot add vectors of two di↵erent sizes, we cannot add a scalar anda vector, and we cannot multiply vectors together.

Note on notation: vectors are usually denoted in boldface, like the vector u, whilescalar numbers are denoted in lightface, like a constant c.

Example 1.1. Given u =

42

�, v =

�13

�, c = 3 , find cu and �2u+ 3u.

1

1.3: Vector Equations 2

2. Geometric Interpretation of vectors

We can think of a vector in R2 as a point in 2D space.

Plot u =

42

�, v =

�13

�.

Adding vectors: u+ v Scaling vectors: cu

1.3: Vector Equations 3

Algebraic properties of vectors:

i) u+ v = v + u

ii) (u+ v) +w = v + (u+w)

iii) u+ 0 = 0+ u = u

iv) u+ (�u) = �u+ u = 0

v) c (u+ v) = cv + cu

vi) (c+ d)u = cu+ du

vii) c (du) = (cd)u

viii) 1u = u

3. Linear combinations

Suppose u1, u2, . . . , un are vectors of the same size, and x1, x2, . . . , xn are constants.Then

y = x1 u1 + x2 u2 + . . .+ xn un

is a linear combination of u1, u2, . . . , un.

For example, with u =

42

�, v =

�13

�, the vector �5u + 4v would be a linear

combination of u, v.

This topic leads to a big question we’ll be asking in this chapter: Given the vectorsu1, u2, . . . , un and another vector y, how can we tell if y is a linear combination ofthe u vectors?

1.3: Vector Equations 4

Example 3.1. Determine if

2

474

�3

3

5 is a linear combination of

2

41

�2�5

3

5 and

2

4256

3

5 .

That is, can we find two values for x1 and x2 such that the equation below is true?

x1

2

41

�2�5

3

5+ x2

2

4256

3

5 =

2

474

�3

3

5 .

1.3: Vector Equations 5

To answer the question “is b a linear combination of a1, a2, . . . , an?” we can rewritethis as a vector equation:

x1 a1 + x2 a2 + . . .+ xn an = b.

Can we solve for the xi constants to make this true?

To find the xi constants, we can write this as an augmented matrix2

4" " " "a1 a2 . . . an b# # # #

3

5 ,

where the matrix has the vectors ai as its columns.

This is written more succinctly as⇥a1 a2 . . . an b

⇤

.

Definition 3.1. If a1, a2, . . . , ap are in Rn, then the set of all linear combinations

of a1, a2, . . . , ap is denoted by span{a1, a2, . . . , ap}, and is called the subset of Rn

spanned by a1, a2, . . . , ap. That is, span{a1, a2, . . . , ap} is the collection of all vectors

that can be written in the form

x1 a1 + x2 a2 + . . .+ xp ap

with x1, x2, . . . , xp scalars.

Asking whether a vector b is in span{a1, a2, . . . , ap} amounts to asking whether thevector equation

x1 a1 + x2 a2 + . . .+ xn an = b

has a solution, or, equivalently, asking whether the linear system with the augmentedmatrix ⇥

a1 a2 . . . an b⇤

has a solution.

Example 3.2. How can we show that the zero vector 0 is in every span span{a1, a2, . . . , ap}?

1.3: Vector Equations 6

4. Geometric Interpretation of Span

Example 4.1. Let

a1 =

2

41

�23

3

5 , a2 =

2

45

�13�3

3

5 , b =

2

4�381

3

5 .

Then, span{a1, a2} is a plane through the origin in R3. Is b in that plane?