Chapter 1 Flashcards
Def: R^n
R^n = {[x1, x2, …, xn] | x1, x2, …, xn ER}
“n-dimensional Euclidean space”
Elements of Rn called “vectors”
Def: vector addition
Let x, y ER^n
x + y = [x1+y1, …, xn+yn] if x=[x1, …, xn] and y=[y1, …, yn]
Def: vector scalar multiplication
Let x ER^n, cER
cx=[cx1, …, cxn] if x=[x1, …, xn]
Def: linear combination
c1v1 + c2v2 + … + ckvk
c1, …, ck ER
v1, …, vL vectors in R^n
Def: Span
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
Theorem 1.2.1 (first theorem)
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}
10 vector axioms
- x+y ER^n
- (x+y)+w = x+(y+w) (associative)
- x+y=y+x (commutative)
- 0ER^n such that x+0=x for all xER^n
- x+(-x) = 0
- cxER^n
- c(dx) = (cd)x
- (c+d)x = cx + dx
- c(x+y) = cx + cy
- 1x = x
Def: linearly dependent/independent
c1v1 + … + ckvk = 0
Linearly dependent: c1, …, ck == 0
Linearly independent: c1, …, ck = 0
Also called the TRIVIAL solution
Def: k-plane or k-flat
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
Theorem 1.2.2 (consequence of first theorem)
A set of vectors {v1, …, vk} in R^n is linearly dependent iff viESpan{v1, …, vi-1, vi+1, …, vk} for some i, 1
Theorem 1.2.3
If a set contains the zero vector, it is linearly dependent
Def: basis
B is a basis for V iff
1) Span(B) = V
2) B is linearly independent
Def: elementary basis vectors
{e1, …, en}
e1 = (1,0,….0)
en = (0,…,1)
They’re called the “standard basis”
Theorem 1.2.4
If B is a basis for V, every vector xEV can be uniquely expressed as a linear combination of the vectors in B
Def: line in R^n
L = {cv + b ER^n | cER} is a line through b in direction of v
Def: sub space of R^n
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
Theorem 1.3.1 (subspace test)
Let S be a nonempty subset of V
If S is closed under addition and scalar multiplication, then S is a subspace
Theorem 1.3.2
Let B = {v1, ..., vk} CV
Then Span(B) is a subspace of V
Reverse is also trueSubspace - trivial case
R^2 is a subspace of R^2
Def: vector space
A vector space (over R) is a set V together with two binary operators: “addition” and “scalar multiplication”
Satisfies 10 axioms
Def: dot/inner product
x•y = x1y1 + … + xnyn
OR
x•y = ||x||•||y||•cosθ
Theorem 1.4.2 (properties of dot product)
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)
Def: length/norm
||x||
Theorem 1.4.3 (properties of length)
1) ||x|| >_0, ||x|| = 0 iff x=0
2) ||cx|| = |c|•||x||
3) |x•y|
Def: orthogonal
If dot product = 0
Def: unit vector
Length = 1
Def: orthonormal
If B is orthogonal and normal
Def: normal
B is normal (unit-length) if ||vi|| = 1 for every i=1,…,k
Def: cross product
R^3
vxw = (v2w3-v3w2, v3w1-v1w3, v1w2-v2w1)
Theorem 1.4.5 (properties of cross product)
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
Theorem 1.4.6 (scalar equation)
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
Def: normal vector and scalar equation
n1x1 + … + n3x3 = b1n1 + … + b3n3
n = (n1, …, n3) is the normal vector
And that equation is the scalar equation
Def: projection
u onto v
projv(u) = u•v/||v||^2 (v)
Def: perpendicular of a projection
Perpendicular of x onto v
Perpv(x) = x-projv(x)
Def: projection into a plane
x onto P
projP(x) = x-perpn(x)
Def: perpendicular of a projection onto a plane
x onto P
perpP(x) = projn(x)
Def: determinant
Given square matrix [ a b / c d ] E M2x2(R), it’s determinant is det[ab/cd] = ad-bc
Properties of length continued
3) triangle inequality
4) |x•y| LESS THAN MAGNITUDE X TIMES MAGNITUDE Y I STG
