Midterm Review Flashcards
Vector Addition
For x = [x1, x2] and y = [y1, y2]; x + y = [x1+y1, x2+y2]
Scalar Multiplication
If t is a real # (scalar), then tx = [tx1, tx2]
Linear Combination
Given any # of vectors, a lin. comb. is sum of scalar multiples of the given vectors. Any vector x = [x1, x2] can be written as x = x1[1, 0] + x2[0, 1]
Zero Vector
[0, 0] , denoted 0 with length but no direction
Vector Equation of a Line Through 0
x = td, teR. The set {td : teR} can be used to describe points on the line.
Vector Equation of a Line not Through 0
Find/Use a point on the line and get a vector p whose head is at that point, and add p to any td with teR.
Equation: x = p + td, teR.
Find the Equation of a Line through (0, 0) parallel to Existing Line?
Use the same direction vector td.
x = td, teR
Parametric Equation
If x = [x1, x2] = [4, -3] + t[3, -1] then the following is the pe
x1 = 4 +3t
x2 = -3 -t
Scalar Equation
From parametric equ solve all equations for the scalar (eg. t) and equate then solve for the form ax1 + bx2 +… = d
Directed Line Segments
Free vectors which are allowed to be translated parallel to themselves.
Parallel Vectors
u and v are said to be parallel if they are scalar multiples of each other. 0 = 0u but u isn’t a sm of 0 yet 0 and u are parallel.
Properties of Vectors in R^n
For any x, y, w eRn and s, t eR: x + y eRn x + y = y + x (x + y) + w = x + (y + w) tx eRn (st)x = s(tx) (s + t)x = sx + tx t(x + y) = tx + ty
Subspaces
A subspace is a non empty subset (say s) of Rn such that for any x ,y eS and teR: x + y eS (closed under addition) and tx eS (closed under scalar multipllication). Note: 0 is always in any subspace.
Proving Subspaces
Closed under addition, closed under scalar multiplication, and non empty. In other wods, check if 0 is eR, if the sum of 2 vectors (which are eS) is also eS, and if the product of a scalar t and a vector (eS) is also eS.
Closed under addition
Let [x1, x2] and [y, y2] be from S. Then [x1, x2] [y1, y2] = [x1+x2, x2+y2]. Then x2 + y2 = (2x1) + (2y1) = 2(x1+y1) Therefore S is closed under addition.
Closed Under Scalar Multiplicaton
Let teR and [x1, x2]eS. Then t[x1, x2] = [tx1, tx2]. Now tx2 = t(2x1) = 2(tx1) therefore S is closed under sm
Non Empty
0 = [0, 0]. As 2(0) = 0, we get 0eS. Therefore S is non empty.
Relation Between Subspces and Subsets
Every subspace is a subset but not every subset is a subspace.
Spanning Sets
If S = {t1v1 + tkvk : t1,…, tk eR} then S is the subspace spanned by B = {v1,…,vk}. This can be said that the set B spans S, or that B is the spanning set for the subspace S.
Linear Dependent
If there exists coefficients t1,…, tk that are not all 0 such that 0 = t1v1+…+tkvk then a set of vectors {v1,…,vk} is linearly dependent
Linear Independent
If the only solution to 0= t1v1+…+tkvk is if all coefficients are 0, then a set f vectors is linearly independent.
Dot Product
If x and y are from Rn then x . y = x1y1 + … + xnyn
Properties of the Dot Product
x . x > 0 and x . x = 0 IFF x = 0
x . y = y . x
x . (y + z) = x . y + x . z
x . (ty) = t(x.y)
Magnitude of x
sqrt( x . x)
Magnitude Properties
mag(tx) = abs(t)mag(x)
abs(x . y) <= mag(x)*mag(y) and = IFF {x,y} is LD
mag(x+y) <= mag(x) + mag(y)
Dot Product with Angles
x . y = mag(x)mag(y)cos(theta)
Planes in R3
If p = [p1,p2,p3] is fixed x = [x1,x2,x3] is any point on p with a normal vector n = [n1,n2,n3], then n . p = n1p1+n2p2+n3p3 = d is the scalar equation of the plane. Ex. 3x1-2x2+x3=5, n=[3,-2,1]
Parallel and Perpendicular Planes
2 planes are parallel if their n’s are parallel
2 planes are perpendicular if their n’s are orthogonal
Projections
proj[y]x = ( (x.y) / (y.y) )*y mag(x)*cos(theta) = (x.y) / mag(y)
Properties of Projections
proj[y]tx = tproj[y]x proj[y](x+z) = proj[y]x + proj[y]z
Perpendicular of a Projection
perp[y]x = x - proj[y]x
Minimum Distance
If a vector x provides us with the coordinates of a point and y is a direction vector for a line through 0, then mag(perp[y]x) will be the minimum distance between the point and the line.
Finding the point on a line closest to a given point p
Need a vector which = another vector q + proj
Finding a distance of a point P from a plane
Find a point q on the plane then project qp onto n and take the mag of the result.
Cross Product
2312x2312
Right Hand Rule for Cross Product
Shows direction of orthogonal vector.
Length: mag(X x Y) = mag(x)mag(y)abs(sin(theta))
Area of a Parellologram
mag(x)mag(y)abs(sin(theta)) which is = mag(XxY)
Volume of a Parallelepiped
Triple Scalar Product = z . (XxY)
REF and RREF
If result contains a zero row, there is a free variable x[n] where n is the column in which there is no leading non-zero entry
Rank of a Matrix
The total number of pivots (leading 1s) in the RREF
Inconsistent Augmented Matrix
IFF the rank of A < the rank of [A l b]
Homogeneous Systems
RHS is always 0 so no need to write augmented matrix, row reduce with coefficient matrix. x = 0 will always be a solution.
Properties of Augmented Matrices
Is consistent IFF rank A = rank [A l b]
IF consistent, then the # of params in the general solution is = n - rankA where n is the number of variables in the augmented matrix
Transpose of a Matrix A[mxn]
A^T = A[nxm]. In other words, the rows become the columns and columns become rows
Matrix Multiplication
Row x column
