Midterm Study Cards

Question 1

What is a linear combination?

Answer 1

The linear combination of two vectors v and w is av + bw where a and b are scalars

Question 2

Are Point and Vector equivalent in linear algebra?

Answer 2

Yes

Question 3

What is the symbol for and the formula for the length/magnitude of a n-vector v

Answer 3

The symbol and equation is
||v||, sqrt((v_1)^2+(v_2)^2+(v_n)^2)

Question 4

How do you prove two vectors are linearly independent?

Answer 4

If there is no way to form a linear combination of them equal to 0 without a and b being 0

Question 5

What is the equation of the distance between two vectors v and w?

Answer 5

The equation is
||v - w||

Question 6

What is the zero vector?

Answer 6

The zero vector in R^n with coordinates [0,0,0,0,…,0]

Question 7

What is a unit vector?

Answer 7

A unit vector is a vector with length 1

Question 8

When are vectors x and y perpendicular

Answer 8

When the dot product of x and y is 0

Question 9

What is the cosine of the angle between two vectors v and w?

Answer 9

The cosine is
(v dot w)/(||v||*||w||)

Question 10

When are two vectors orthogonal?

Answer 10

When they are perpendicular to each other

Question 11

What 5 rules hold up for all vectors?

Answer 11

v dot w = w dot v
v dot v = ||v||^2
v dot (cw) = c(v dot w) (where c is a scalar)
v dot (w1 + w2) = (v dot w1) + (v dot w2)
v dot (c1w1 + c2w2) = c1(v dot w1) + c2(v dot w2)

Question 12

What is the correlation coefficient between two vectors v and w?

Answer 12

The equation is
(v dot w)/(||v||*||w||) where v_avg and w_avg = 0

Question 13

What does a correlation coefficient close to -1 imply?

Answer 13

The data points are close to a line of negative slope

Question 14

What does a correlation coefficient close to 1 imply?

Answer 14

The data points are close to a line with positive slope

Question 15

What does a correlation coefficient close to 0 imply?

Answer 15

x and y do not have a strong linear correlation

Question 16

What is the basic equation of a plane?

Answer 16

ax + by + cz = d where a, b, c, and d are constants

Question 17

What is the parametric form of the equation of a plane?

Answer 17

P + te + t’e’ where t and t’ are all possible scalar values

Question 18

What is the normal vector form of the equation of a plane?

Answer 18

A plane is defined by stating it is perpendicular to the vector n and passes through the point w.

Question 19

How are the normal vector n to the plane W and the plane W which passes through the orgin related?

Answer 19

For W = ax + by + cz = 0, n = [a, b, c] for R^3. This applies to all R^n

Question 20

What is the three point form of the equation of a plane?

Answer 20

A plane described by 3-points which are NOT collinear

Question 21

What is the span of vectors v1, v2, …, vk in R^n

Answer 21

The span of the vectors is all possible places in the space R^n that can be reached via a linear combination of the vectors
Or
span(v1,…,vk) = {all n-vectors x of the form c1v1,…,ckvk}

Question 22

What must be true of spans?

Answer 22

They pass through the origin

Question 23

What is a linear subspace of R^n?

Answer 23

A linear subspace of R^n is a subset of R^n that is the span of a finite collection of vectors in R^n

Question 24

Given a linear subspace V, what are true of the vectors within it?

Answer 24

For all vectors that exist in V, any linear combination of those vectors exist in V (aka they belong to the same linear subspace)

Question 25

What is Dim(V)

Answer 25

Dim V is the dimension of V aka the smallest number of vectors needed to span V.

Question 26

If W and V are both linear sub spaces in R^n and W is contained in V, what is true of W

Answer 26

W = V

Question 27

What is a basis

Answer 27

A basis is a set of n vectors which spans a linear subspace of dim = n

Question 28

When is a collection of vectors considered orthogonal?

Answer 28

When for vi,…,vk, vi dot vj = 0 when i!=j

Question 29

When is a collection of vectors orthonormal?

Answer 29

A collection of vectors is orthonormal if they’re all orthogonal and unit vectors

Question 30

How can you convert orthogonal basis’ to orthonormal basis’?

Answer 30

Divide each vector by it’s length

Question 31

What does the Fourier formula do and what is it?

Answer 31

The Fourier formula can be used to get the scalar multiples which create the vector v which lies on the span of a collection of orthogonal vectors, v = summation from 1 to k of ((v dot v_i)/(vi dot vi))vi where the collection of orthogonal vectors has the form {vi,…,vk}.

Question 32

What is the Projection and what’s it’s formula?

Answer 32

The Projection is the closest point from one vector to another, the formula for Proj_w(x) = ((x dot w)/(w dot w))w

Question 33

Given a collection of vectors which make up a linear subspace, how do you find the point closest to a point on that linear subspace?

Answer 33

Let V be the linear subspace = {v1,v2,…,vk} then Proj_V(x) = Proj_v1(x) + Proj_v2(x) + … + Proj_vk(x)

Question 34

If V is a linear subspace of R^n how can every vector x in R^n be uniquely expressed?

Answer 34

x = v + v’ where v = Proj_V(x) and v’ = x - Proj_V(x)

Question 35

How can you find the orthogonal basis of the span(x,y)

Answer 35

The orthogonal basis would be y and x’ = x - Proj_y(x)

Question 36

How do you find the line of best fit for a data set?

Answer 36

Break the points into vector X with all the x values and vector Y with all the y vectors. Then calculate x_avg and y_avg. Create a vector X’ = [x1-x_avg, x2 - x_avg, …, xk- x_avg] Then these into the formula Proj_X’(Y) + Proj_1(Y) the coefficients of these vectors are the equation of the line. Note that X’ = X - x_avg([1])

Question 37

What is the value of g(f(x)) for a Multivariable problem

Answer 37

G applied to F. So f’s output values plugged into G.

Question 38

What is a saddle point?

Answer 38

A saddle point is a point where both f_x and f_y are 0 but while in on direction it appears the point is a minimum, but the other it appears to be a maximum

Question 39

How do you calculate the linear approximation of a function f?

Answer 39

For x near a ∈ Rn , the linear approximation to f is f(x) ≈ f(a) + ((∇f)(a)) · (x − a).

Question 40

What is the equation of the subspace tangent to the curve on a contour plot?

Answer 40

At the point a, b the tangent vector is GradF(a,b) dot [x - a, y - b] = 0

Question 41

How can you immediately tell if a set of vectors is not linearly independent?

Answer 41

If it has a vector which is not non-zero

Question 42

When is a vector non-zero?

Answer 42

When it has at least one entry which is non-zero

Question 43

Upper left to lower right diagonal of inverse square matrix is always what?

Answer 43

1/The original value of the position in the diagonal

Question 44

REVIEW 11/29/2022 LECTURE (Specifically for inverse matrix review) Also reveal about midterm potentially at around 30 minutes in (go like 24 and watch should be on the slide with blue box on the top also around 52 (go to like 47)

Answer 44

YES

Question 45

What is the determinant of an upper or lower triangular matrix?

Answer 45

The product of it's diagonal entries