Chapter 1

Question 1

Def: R^n

Answer 1

R^n = {[x1, x2, …, xn] | x1, x2, …, xn ER}
“n-dimensional Euclidean space”
Elements of Rn called “vectors”

Question 2

Def: vector addition

Answer 2

Let x, y ER^n

x + y = [x1+y1, …, xn+yn] if x=[x1, …, xn] and y=[y1, …, yn]

Question 3

Def: vector scalar multiplication

Answer 3

Let x ER^n, cER

cx=[cx1, …, cxn] if x=[x1, …, xn]

Question 4

Def: linear combination

Answer 4

c1v1 + c2v2 + … + ckvk
c1, …, ck ER
v1, …, vL vectors in R^n

Question 5

Def: Span

Answer 5

Let B = {v1, v2, …, vk} ER^n
Span(B) = {c1v1 + … + ckvk | c1, …, ck ER}

Set Span(B) is SPANNED by B
B is a SPANNING SET for Span(B)

Span is all linear combinations

Question 6

Theorem 1.2.1 (first theorem)

Answer 6

Let v1, …, vk ER^n
Some vector vi, 1= i = k can be written as a linear combination of v1, …, vi-1, vi+1, …, vk iff
Span{v1, …, vk} = span{v1, .., vi-1, vi+1, …, vk}

Question 7

10 vector axioms

Answer 7

1. x+y ER^n
2. (x+y)+w = x+(y+w) (associative)
3. x+y=y+x (commutative)
4. 0ER^n such that x+0=x for all xER^n
5. x+(-x) = 0
6. cxER^n
7. c(dx) = (cd)x
8. (c+d)x = cx + dx
9. c(x+y) = cx + cy
10. 1x = x

Question 8

Def: linearly dependent/independent

Answer 8

c1v1 + … + ckvk = 0
Linearly dependent: c1, …, ck == 0
Linearly independent: c1, …, ck = 0
Also called the TRIVIAL solution

Question 9

Def: k-plane or k-flat

Answer 9

Let V = R^n, B = {v1, ..., vk} CR^n
If B is linearly independent, and bER^n, 
P=b+Span(B)
So P={b+c1v1+...+ckvk | c1, ..., ck ER}
Called a k-dimensional plane of R^n
Aka k-plane or k-flat

k=2: plane in R^n
k=n-1: hyperplane in R^n

Question 10

Theorem 1.2.2 (consequence of first theorem)

Answer 10

A set of vectors {v1, …, vk} in R^n is linearly dependent iff viESpan{v1, …, vi-1, vi+1, …, vk} for some i, 1

Question 11

Theorem 1.2.3

Answer 11

If a set contains the zero vector, it is linearly dependent

Question 12

Def: basis

Answer 12

B is a basis for V iff

1) Span(B) = V
2) B is linearly independent

Question 13

Def: elementary basis vectors

Answer 13

{e1, …, en}
e1 = (1,0,….0)
en = (0,…,1)

They’re called the “standard basis”

Question 14

Theorem 1.2.4

Answer 14

If B is a basis for V, every vector xEV can be uniquely expressed as a linear combination of the vectors in B

Question 15

Def: line in R^n

Answer 15

L = {cv + b ER^n | cER} is a line through b in direction of v

Question 16

Def: sub space of R^n

Answer 16

Suppose a subset ScV satisfies all 10 axioms of vector space using the same 2 operations inherited from V
Then we say that the vector space S is a subspace of V

Question 17

Theorem 1.3.1 (subspace test)

Answer 17

Let S be a nonempty subset of V

If S is closed under addition and scalar multiplication, then S is a subspace

Question 18

Theorem 1.3.2

Answer 18

Let B = {v1, ..., vk} CV
Then Span(B) is a subspace of V
Reverse is also true

Question 19

Subspace - trivial case

Answer 19

R^2 is a subspace of R^2

Question 20

Def: vector space

Answer 20

A vector space (over R) is a set V together with two binary operators: “addition” and “scalar multiplication”
Satisfies 10 axioms

Question 21

Def: dot/inner product

Answer 21

x•y = x1y1 + … + xnyn
OR
x•y = ||x||•||y||•cosθ

Question 22

Theorem 1.4.2 (properties of dot product)

Answer 22

1) x•y >_0, x•x=0 iff x=0
2) x•y = y•x
3) x•(sy+tz) = s(x•y) + t(x•z)

Question 23

Def: length/norm

Answer 23

||x||

Question 24

Theorem 1.4.3 (properties of length)

Answer 24

1) ||x|| >_0, ||x|| = 0 iff x=0
2) ||cx|| = |c|•||x||
3) |x•y|

Question 25

Def: orthogonal

Answer 25

If dot product = 0

Question 26

Def: unit vector

Answer 26

Length = 1

Question 27

Def: orthonormal

Answer 27

If B is orthogonal and normal

Question 28

Def: normal

Answer 28

B is normal (unit-length) if ||vi|| = 1 for every i=1,…,k

Question 29

Def: cross product

Answer 29

R^3

vxw = (v2w3-v3w2, v3w1-v1w3, v1w2-v2w1)

Question 30

Theorem 1.4.5 (properties of cross product)

Answer 30

1) wxv = -vxw
2) vxv = 0
3) vx(w+x) = vxw + vxx
4) vx(cw) = (cv) x w = c(vxw)
5) if n=vxw, then n is orthogonal to the subspace Span{v,w} (ie if yESpan{v,w} then n•y=(vxw)•y=0)
6) vxw=0 iff {v,w} is linearly dependent
7) ||vxw|| = ||v||•||w||•sinθ (0

Question 31

Theorem 1.4.6 (scalar equation)

Answer 31

Let P = b+Span{v,w} be a 2D plane in R^3
Let n = vxw
P = {xER^3 | n•x = n•b} = (x-b)•n = 0

Question 32

Def: normal vector and scalar equation

Answer 32

n1x1 + … + n3x3 = b1n1 + … + b3n3

n = (n1, …, n3) is the normal vector
And that equation is the scalar equation

Question 33

Def: projection

Answer 33

u onto v

projv(u) = u•v/||v||^2 (v)

Question 34

Def: perpendicular of a projection

Answer 34

Perpendicular of x onto v

Perpv(x) = x-projv(x)

Question 35

Def: projection into a plane

Answer 35

x onto P

projP(x) = x-perpn(x)

Question 36

Def: perpendicular of a projection onto a plane

Answer 36

x onto P

perpP(x) = projn(x)

Question 37

Def: determinant

Answer 37

Given square matrix [ a b / c d ] E M2x2(R), it’s determinant is det[ab/cd] = ad-bc

Question 38

Properties of length continued

Answer 38

3) triangle inequality

4) |x•y| LESS THAN MAGNITUDE X TIMES MAGNITUDE Y I STG