Linear Algebra 1 Revision Session!

Complex Numbers

100

a = 2 + i

b = -3 - 2i

Find a + b 

-1 - i

200

a = 2cis(30)

b = 3cis(20)

Find a x b

6cis(50)

300

Convert the following to polar form:

2 + i

√5 cis(26.57)

400

What is the centre and radius of the following complex region:

|z + 4| > 3

Centre at (-4, 0)

Radius = 3

500

Draw the following complex region:

Im(z) < 2

y < 2 

Area below line y = 2

Vectors

100

Find the dot/scalar product between:

[1, 2, 1] and [3, 6, 8]

23

200

Find the unit vector of:

[2, 1, 3] 

[2/4, 1/4, 3/4] 

300

Find the cross product between:

[0, 1, 1] and [2, 1, 0]

[-1, 2, -2]

400

Find the angle between:

[3, 1, 2] and [-2, 2, 2]

0

500

Find the vector projection of a on b

a = [1, 3] and b = -2i + 2j

[-1, 1]

Matrices

100

Find 3A if:

A = [2 3 -6]

[6 9 -18]

200

Find the transpose of the following matrix:

[2 4 6 

 3 2 5

 9 1 8]

[2 3 9 

 4 2 1 

 6 5 8] 

300

Find the inverse of the following matrix:

-1 2

-1 4

1/2 [4  -2

       1 -1]

400

Find vector in the same direction as c but with a length of 4 units.

𝒄 = [2, −2,1]

[8/3, -8/3, 4/3]

500

Multiply the following matrices:

[2 1           [4 6

 3 5]   and   2 3]

10 15

22 33

Systems of Linear Equations

100

Find the determinant of the following matrix:

[3 4

6  8]

0

200

How many solutions does the following system of linear equations have?

[1 0 2| 1

 0 2 3| 2

 0 0 0| 0] 

Infinite solutions

300

If a matrix A is 6 × 4 and the product AB is 6 × 8, what is the order (dimensions) of B?

B = n × m = 4 × 8

400

Find C² if C=

[1 2

-2 1]

-3 4

-4 -3

500

Consider the following systems of equations,

3x₁ + x₂ = 11

−2x₁ + 3x₂ = −22

Using Cramer’s Rule determine the values of x₁

x₁ = 5

Euclidean Vector Spaces

100

Find the magnitude of the following vector in R⁶

[-2, 3, 0, -5, -7, 7]

2√34

200

Given the plane, (𝑥 + 2) + 4(𝑦 − 2) − 2𝑧 = 0

Does the point (2, 1, 0) lie on the plane?

Yes

300

Find the vector equation of the line passing through the points (−4,0,2) and (−1,3,5)

r = [-4, 0 , 2] +t[3, 3, 3]

400

Are the following pair of lines parallel, perpendicular or skew?

−8x − 6y + 2z = 1

z = 4x + 3y

Parallel

500

Find the angle between the planes 

x + y + z = 0 and x + 2y + 3z = 1.

θ = cos^-1(6/√ 3 x √ 14)? = 22.21^o