Matrix Midterm 1

cross product formula

basis + how to solve for it

A basis in V, is a set of vectors that
1. Span V ( every vector in V can be written as linear combination of basis)

2.Are linearly IND

subspace + how to solve for it

Subspace of Rn is a set of vectors that:
1. Contain the zero vector

2. are closed under addition and scalar multiplication

span + how to solve for it

The span of a set of vectors is all possible linear combination of those vectors ( written out in generic format)

projection of x onto y

where y dot y is the same thing as the lenght of y squared

perpendicular of x onto y

Nearest Point formula

if the nearest point to Q we’re solving for we call R.

OR = OQ + Projn (QP) , where n is the normal to the line/plane and P is a point on the line/plane

Min Distance in R2

lenght of perp(d)(PQ) perpendicular of directed line segment PQ onto d ( d is subscript), where P is point ur trying to reach and Q is some point on line?

Min Distance in R3

lenght of projection of PQ onto the normal ( normal is subscript)- where Q is point ur trying to reach and P is some point on the line

Volume of Parralelpiped

if u,v,and w are adjacent edges of parrelpiped

the volume is the absolute value of:

((U x V) (dot)W or the det of this 3×3 matrix )

Scalar Tripe Product

(UxV) dot W → any combo of these vectors works

ORRR the det of these vectors if we arrange them into a 3×3 matrix?

ERO notation

RREF FORM

• matrix is in row echelon form

• all leading values are 1

• in a coloumn with a leading one, all other values are 0

• ( coloumns with 1’s that are not leading can still have other numbers above/below)

Column of zeros + free varibles( key reminder)

• column of zeros does not mean the variable = 0

• Just means that the varible never appears in eqs, so nothing restricts it

• therefore it is a free varible ( u cant write any of the terms except itself in terms of it)

• here if x1 is free varible s, only x1=S

REF form

All nonzero rows are above any rows of all zeros

• The first nonzero entry is the pivot (leading entry does not have to be 1)

• Each pivot is to the right of the pivot in the row above

• All entries below each pivot are zero

acute vs obstuse angle

a cute angle is less than 90 degrees ( or pi/2 radians)

an obtuse ( large) angle is more than 90 degrees

format of a basis ( in answer)

s=span(basis vectors)

steps for success in proofs

• whenever u have more than 1 solution → free varibles ( if u dont have all free varibles, rewrite other varibles in terms of free varibles)

• go with a generilization not plugging in specific answer

• generlization = assigning paramters ( ex. x2=S)

• then rewrite whatever statement/criteria u have as a linear combo of the paramteres ( same way u do when ur writing a general equation for a matrix)

how to find PQ if P and Q are points

OQ-OP ( where O is the origin- should be teh same as Q-P tbh)