Midterm Review

Question 1

Vector Addition

Answer 1

For x = [x1, x2] and y = [y1, y2]; x + y = [x1+y1, x2+y2]

Question 2

Scalar Multiplication

Answer 2

If t is a real # (scalar), then tx = [tx1, tx2]

Question 3

Linear Combination

Answer 3

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]

Question 4

Zero Vector

Answer 4

[0, 0] , denoted 0 with length but no direction

Question 5

Vector Equation of a Line Through 0

Answer 5

x = td, teR. The set {td : teR} can be used to describe points on the line.

Question 6

Vector Equation of a Line not Through 0

Answer 6

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.

Question 7

Find the Equation of a Line through (0, 0) parallel to Existing Line?

Answer 7

Use the same direction vector td.

x = td, teR

Question 8

Parametric Equation

Answer 8

If x = [x1, x2] = [4, -3] + t[3, -1] then the following is the pe
x1 = 4 +3t
x2 = -3 -t

Question 9

Scalar Equation

Answer 9

From parametric equ solve all equations for the scalar (eg. t) and equate then solve for the form ax1 + bx2 +… = d

Question 10

Directed Line Segments

Answer 10

Free vectors which are allowed to be translated parallel to themselves.

Question 11

Parallel Vectors

Answer 11

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.

Question 12

Properties of Vectors in R^n

Answer 12

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

Question 13

Subspaces

Answer 13

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.

Question 14

Proving Subspaces

Answer 14

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.

Question 15

Closed under addition

Answer 15

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.

Question 16

Closed Under Scalar Multiplicaton

Answer 16

Let teR and [x1, x2]eS. Then t[x1, x2] = [tx1, tx2]. Now tx2 = t(2x1) = 2(tx1) therefore S is closed under sm

Question 17

Non Empty

Answer 17

0 = [0, 0]. As 2(0) = 0, we get 0eS. Therefore S is non empty.

Question 18

Relation Between Subspces and Subsets

Answer 18

Every subspace is a subset but not every subset is a subspace.

Question 19

Spanning Sets

Answer 19

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.

Question 20

Linear Dependent

Answer 20

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

Question 21

Linear Independent

Answer 21

If the only solution to 0= t1v1+…+tkvk is if all coefficients are 0, then a set f vectors is linearly independent.

Question 22

Dot Product

Answer 22

If x and y are from Rn then x . y = x1y1 + … + xnyn

Question 23

Properties of the Dot Product

Answer 23

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)

Question 24

Magnitude of x

Answer 24

sqrt( x . x)

Question 25

Magnitude Properties

Answer 25

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)

Question 26

Dot Product with Angles

Answer 26

x . y = mag(x)mag(y)cos(theta)

Question 27

Planes in R3

Answer 27

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]

Question 28

Parallel and Perpendicular Planes

Answer 28

2 planes are parallel if their n’s are parallel

2 planes are perpendicular if their n’s are orthogonal

Question 29

Projections

Answer 29

proj[y]x = ( (x.y) / (y.y) )*y
mag(x)*cos(theta) = (x.y) / mag(y)

Question 30

Properties of Projections

Answer 30

proj[y]tx = tproj[y]x
proj[y](x+z) = proj[y]x + proj[y]z

Question 31

Perpendicular of a Projection

Answer 31

perp[y]x = x - proj[y]x

Question 32

Minimum Distance

Answer 32

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.

Question 33

Finding the point on a line closest to a given point p

Answer 33

Need a vector which = another vector q + proj

Question 34

Finding a distance of a point P from a plane

Answer 34

Find a point q on the plane then project qp onto n and take the mag of the result.

Question 35

Cross Product

Answer 35

2312x2312

Question 36

Right Hand Rule for Cross Product

Answer 36

Shows direction of orthogonal vector.

Length: mag(X x Y) = mag(x)mag(y)abs(sin(theta))

Question 37

Area of a Parellologram

Answer 37

mag(x)mag(y)abs(sin(theta)) which is = mag(XxY)

Question 38

Volume of a Parallelepiped

Answer 38

Triple Scalar Product = z . (XxY)

Question 39

REF and RREF

Answer 39

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

Question 40

Rank of a Matrix

Answer 40

The total number of pivots (leading 1s) in the RREF

Question 41

Inconsistent Augmented Matrix

Answer 41

IFF the rank of A < the rank of [A l b]

Question 42

Homogeneous Systems

Answer 42

RHS is always 0 so no need to write augmented matrix, row reduce with coefficient matrix. x = 0 will always be a solution.

Question 43

Properties of Augmented Matrices

Answer 43

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

Question 44

Transpose of a Matrix A[mxn]

Answer 44

A^T = A[nxm]. In other words, the rows become the columns and columns become rows

Question 45

Matrix Multiplication

Answer 45

Row x column