1) Determinants

Question 1

Area and determinants

Dot product

Answer 1

SIGNED area of parallelogram with col vectors,
•Sign depends on sin
•positive if 0-pi

Area(r,r’) = det(r,r’) = magnitudes * sin( angle)
= dot product of (a,c) and (d,-b)

Also works with rows: det A^T = det A

Question 2

Vector orthogonal to (b,d)

Answer 2

(b,d) • (d,-b) = 0

Question 3

Linear independence

Answer 3

For vectors v_1, … ,v_n to be linearly independent st for t_1,…t_2 in reals
We have
t_1(v_1) +…+ t_n(v_n) =0

Only for t_1=…=t_2=0

• Some may be 0: such as for n=2 consider CASES of each not equal to zero
• must show that ALL are 0
• a set of vectors is linearly dependent if at least two are multiples- anti-parallel or parallel

Question 4

If vectors are linearly independent then..
Area
Det
LINEAR MAP

THEOREM FOR EQUIVALENCE

Answer 4

Equivalent for 2x2:

Det A≉ 0 EQUIVALENT
Columns are linearly INDEPENDENT EQUIVALENT
ROWS are LINEARLY INDEPENDENTEQUIVALENT
Linear map L: R² ->R² is a BIJECTION

Equivalent for 3X3:

Det A≉ 0

EQUIVALENT
Columns are linearly INDEPENDENT
EQUIVALENT

Linear map L: R³ ->R³ is INJECTIVE

Question 5

Cross products : 3x3 only

Answer 5

Orthogonal to each vector
u X v =
(u_2v_3+ …,…,…)

u X v = v X u

u• ( u X v) = 0
v• (u X v) = 0

Take out scalers, put back in

Question 6

Triple scalar product

Answer 6

Equiv to determinant:
So =0 if linearly dependent
Linearly independent if and only if
Triple scalar not equal to 0

u • (v X w)
= det(u,v,w)
Cyclic property

SIGNED VOLUME:
so absolute value gives actual volume
Hence matrix scales det scaled volume and area

Question 7

Spherical polar coords

Answer 7

Given spherical polar
x=rsinϴcosφ
y=rsinϴsinφ
z=rsinϴcosφ

ϴ from positive z axis to (x,y,z) between 0 & pi
φ from positive x-axis to (x,t,0) between 0 & 2pi

Derivative matrix -> vectors are linearly independent if and only if sin theta not equal to 0 i.e. ϴ not equal to kpi

Jacobian is partial(x,y,z) = r^2 sinϴ
Partial(r,ϴ,φ)

Question 8

Linearly indep and cross products

Answer 8

Linearly independent if and only if
r_1 X r_2 not equal to 0 vector

Shown by
If dependent can be written in terms,
u X u =0 vector
By cases t_1 not equal to zero

Question 9

Proof for linear map is injective when det A is not equal 0

Answer 9

Show injective as properties v-v’ =0

V=v’

Question 10

4x4 plus… we use determinants not cross products

Answer 10

Working in rows determinants to see if linear independent etc

Question 11

Theorem for determinant function

Answer 11

Theorem: there is exactly one function F:R^n X … X R^n -> R
( n times)
With properties: for any 1 less than or equal to I less than or equal to j less than or equal to n

1) swapping row swaps sign
F(r
2) split into addition of two terms
F(r_1 + r'_1, r_2,...,r_n) = F(r_1,...,r_n) + F(r'_1, r_2,...,r_n)
3) take out scalar of ONE ROW
4) F of standard bases = 1

Combining these allows to add multiples of rows which doesn’t change
One repeated row implies F=-F implies 0

Question 12

For n by n matrix det(AB)

Answer 12

Det(AB) detA • detB

Corresponds to linear map and also scales area and volume
Columns:

Area ( L(u) ,L(v)) = det A area(u,v)

Columns u,v now L(u) and L(v)

Similarly volume