Wk 2: The Vector Space

Question 1

Linear combination of vectors

Answer 1

α₁v₁ + α₂v₂ + … + α_nv_n

α₁,…,α_n are coefficients of the linear combination

Question 2

Trivial linear combination

Answer 2

Coefficients are all zero

Question 3

Span of v₁, …, v_n

Answer 3

Set of all linear combinations of vectors v₁, …, v_n.

Written Span {v₁, …, v_n}

Question 4

How many vectors in Span {}

Answer 4

One: the zero vector

Question 5

Generating set

Answer 5

Let

Question 6

Linear combination of linear combinations of vectors

Answer 6

Can be converted to a linear combination of new vectors

Question 7

Standard generators for ℝⁿ

Answer 7

e₁, e₂, …, e_n = [1,0,…,0], [0,1,0,…,0], …, [0,…,0,1]

Question 8

Span of a nonzero vector over ℝ

Answer 8

Span {v} = {αv : α ∈ ℝ}

Question 9

Is the span of k vectors always k-dimensional

Answer 9

No.

eg. Span{[0,0]} is 0-dimensional

Question 10

Specification of a plane

Answer 10

{(x, y, z) : ax + by + cz = 0} OR

{[x, y, z] : [a, b, c] · [x, y, z] = 0}

Question 11

Two ways to represent a geometric object containing the origin:

Answer 11

• Span of some vectors
• Solution of some system of linear equations with zero right-hand sides

Question 12

Vector space

Answer 12

Any subset

Question 13

Subspace

Answer 13

If

Question 14

Every subspace of ℝ^D can be written as:

Answer 14

Span {v₁, . . . , v_n} OR

{x : a₁ · x = 0, …, a_m · x = 0}

Question 15

Convex combination

Answer 15

A linear combination in which all coefficients are nonnegative and sum to 1

Question 16

Convex hull

Answer 16

of a single vector: a point

of two vectors: a line segment

of three vectors: a triangle

Question 17

How to represent a vector space that does not contain the origin

Answer 17

Translate a vector space

Question 18

Affine space

Answer 18

If c is a vector and

Question 19

Affine combination

Answer 19

a linear combination α₁u₁ + α₂u₂ + … + α_nu_n where α₁ + α₂ + … + α_n = 1

Question 20

Affine hull

Answer 20

the set of all affine combinations of a set of vectors

Affine hull of u₁, u₂, . . . , u_n = u₁ + Span {u₂ - u₁, …, u_n - u₁}

Question 21

Thm. Solution set of a linear system

Answer 21

The solution set of a linear system is either empty or an affine space

Question 22

Homogeneous linear equation

Answer 22

A linear equation with a zero right-hand side

α ∙ x = 0

Question 23

Homogeneous linear system

Answer 23

a system of homogeneous linear equations

Question 24

Lemma. Finding a second solution to a linear system

Answer 24

Let u₁ be a solution to a linear system. Then, for any other vector u₂, u₂ is also a solution if and only if
u₂-u_{<strong>1</strong>} is a solution to the corresponding homogeneous linear system

Question 25

checksum function

Answer 25

maps long files to short sequences - used to check for file corruption