Linear Algebra Preparation

Projections/Components

100

find the vector component of v= <1,0> onto u =<3,4>

3/sqrt(5)

200

find the vector projection of v= <3,4> onto u =<5,-12>

we'll figure it out together!

300

find the projection of v= <7,2,5> onto u =<4,4,8>

we'll figure it out together!

400

What is the component formula?

(a•b)/(||b||)

500

What is the Projection Formula?

((a•b)/(||b²||))b

Dot Products

100

<1,0> • <0,1>

0

200

<1,3> • <5,3>

14

300

<1,5,6> • <1,9,2>

58

400

<5,7,2,9> • <6,9,5,8>

175

500

Express a • b in generic form

Lots of answers are acceptable here, like summations of ...s with a₀b₀ etc

Decomposing Matrices

100

What behavior does this matrix have:

[[1, 0],

 [0, 1]]

Nothing - its the identity matrix

200

What behavior does this matrix have:

[[5, 0],

 [0, 1]]

Stretches vectors horizontally by a factor of 5

300

What behavior does this matrix have:

[[0, 0],

 [0, 1]]

Projects the vector onto the y-axis

400

What behavior does this matrix have:

[[0, 1],

 [-2, 0]]

90 degree ccw rotation, and scales by 2

500

Give me the generic 2d matrices for scaling (the x and y individually) and reflecting(across the x,y, and x=y)

scaling: [[k_x, 0],

             [0, k_y]]

reflecting: [[1, 0],            [[-1, 0],              [[0, 1],   

                [0, -1]]             [[0, 1]                [1, 0]]

Matrix Operations

100

[[1, 0],          +           [[1, 0],

 [0, 1]]                         [0, 1]]

[[2, 0],

  [0,2]]

200

[[7, v],          +           [[3, 2],

 [4, 5]]                         [4, 1]]

[[10, 2+v],

  [9,       6]]

300

[[2, 0],         multiply by           [[3, 2],

 [0, 3]]                                     [0, 2]]

[[6,4],

 [0,6]]

400

[[2, 5],         multiply by           [[3, v],

 [7, 3]]                                     [6, 2]]

[[36,2v+10],

 [39,7v+6]]

500

Give me your favorite matrix operation and solve the problem we give you :)

we shall decide whether you're worthy

Mystery

100

pick a number from 1-10

you get 100(x-5) points

200

how do you normalize a vector, how do you notate that, and what does the vector symbolize in such a scenario

you divide by the magnitude, you give it a hat(î), and now it just tells you direction.

300

Find the determinant:

[[3, 1],

 [-2, 6]]

20

400

How many points do you want to wager

If the sum of the digits is odd, you lose the points(in addition to the points you would lose for getting the question wrong)

If the sum of the digits is even, you double the points

500

What is the generic 2d rotation matrix

[[cos(θ), -sin(θ)],

 [sin(θ),  cos(θ)]