Vectors
Matrices
Subspaces
Determinants
Eigen-things
100

Find the distance between the vectors

u = [1 0 1]

v = [2 2 2]

sqrt(6)

100

Is the matrix 

[1 2

 5 10]

invertible?

No! Its determinant is 0.

100

Which vector must always be in any subspace of a vector space?

The zero vector

100

Find the determinant of 

[1 2 3

0 4 5

0 4 5]

0

100

Define what it means for a vector to be an eigenvector of a matrix A

There is a nonzero vector x so that

Ax = (lambda)x

for some scalar (lambda)

200

Write [1 2 3] as a linear combination of

u = [1 0 1]

v = [0 1 1]

w = [-1 0 1]

1u + 2v +0w

200

Is the matrix

[1 0 1 2 3

 0 1 2 0 1

 0 0 0 1 0]

in reduced row echelon form?

No!

There is a 2 above the third pivot

200

What is the definition of the Null Space of a matrix A?

The set of solutions to Ax=0.

200

The determinant of

[-1 2 3

0 9 9

0 0 3]

-27

200

What are the eigenvalues of the matrix

[2 -3

 3 2]?

2 + 3i, 2-3i

300

Are the vectors u,v,w linearly independent?

v = [1 2 3]

w = [4 5 6]

u  = [7 8 9]

No! The determinant of 

[1 4 7

 2 5 8

 3 6 9]

is 0, so the columns are not linearly independent.

300

If A is 5 x 3 and B is 3 x 6, what are the dimensions of AB and BA?

AB is 5x6, BA does not exist

300

Is the vector u = [1 2 3]

in the column space of 

[1 0 0

 0 2 1

 1 0 0]?

No! The augmented matrix

[1 0 0 1

 0 2 1 2

 1 0 0 3]

represents a system with no solution.

300

Find the area of the parallelogram with corners given by

(0,0), (1,1), (2,3), (3, 4)

1

300

Find the eigenvalues and eigenvectors of the matrix

[3 0 1

 0 2 3 

 0 0 4]

3, 2, 4

[1,0,0]

[0,1,0]

[1,3/2,1]

400

Consider the basis

u_1 = [1 2 3]

u_2 = [1 0 1]

u_3 = [0 1 0]

If x has coordinate vector [4 3 2] find x.

[7 10 15]

400

Find the inverse of the matrix

[1 0 1

 0 1 0

 0 2 1]

[1 2 -1

 0 1 0

 0 -2 1]

400

Find a basis of the column space of

[1 2 3

 4 5 6

 7 8 9]

and find its dimension.

Basis: [1 4 7], [2 5 8]

Dimension: 2

400

Use Cramer's rule to solve

3x + 3y = 17

x - 2y = 13

x = 73/9

y = -22/9

400

Diagonalize the matrix

[2 1 0

 0 3 0

 0 0 3]

D = diag(2, 3, 3)

P = 

[1 0 1

 0 0 1

 0 1 0]

500
Find the projection of the vector

u = [1 2 3]

onto the line spanned by v
v = [3 2 1]

projection is:

5/7 [3 2 1]

OR

[15/7, 10/7, 5/7]

500
Suppose T is a linear transformation given by 

T(x_1, x_2, x_3) = (x_1 + x_2, x_2 + x_3, x_3).

Is there a vector whose image under T is (1, 1, 1)?

Find that vector if so, or explain why not.

Yes, we find that 

(x_1, x_2, x_3) = (1, 0, 1)

500

Is it possible for the rank of a 5 x 7 matrix A to be 4? If so, what must be true about the null space of A?

Yes! Rank = 4 means A has 4 pivots.

In this case, A has 3 non-pivot columns, so Ax=0 will have 3 free variables, and thus the dimension of the null space is 3 (i.e. the nullity of A is 3).

500

A linear transformation transforms an image by shrinking its area by a factor of 1/4 and not distorting the image in any direction. What is the linear transformation?

T(x) = Ax where A = 

[1/2 0

 0 1/2]

500

The matrix 

[-1 -1

 2 1]

is similar to a matrix with rotates and rescales vectors. Find that matrix and the angle by which it rotates vectors

lambda = i, -i

matrix = 

[0 -1

 1 0]

Rotation by pi/2 radians (90 degrees)