Linear Algebra Review

Misc.

100

What was the mathematician's name in the movie?

John Nash

200

1- Does it matter if the determinant of a matrix is 19 as oppose to -35? 

2- Does any value of the determinant matter?

1- No, not really

2- It matters if it's 0 or not.

300

A course in linear algebra mostly revolves around what single concept?

Vector spaces

400

How is the Fourier Series approximation of a function different from a Taylor Series approximation of a function?

Fourier Series is arrived at through the process of integration and the Taylor Series is arrived at through differentiation.

500

In finding eigenvalues, why do we set the determinant of A-(lambda)I equal to 0?

That matrix is the coefficient matrix for a homogeneous system. In order for a homogeneous system to have a non-trivial solution, we need it to be non-invertible (singular). This directly follows from the equivalency statements.

Systems and Matrices (Ch. 1-3)

100

1.2(29)

x=2a/3 - b/9

y=-a/3 + 2b/9

200

find the inverse of the matrix in 1.5 (10a)

[16, -5, 3, -1]

300

1.9 (1)

(40, -10, 10)

400

2.2 (20)

determinant = (-6)(2) = -12

500

3.5 (8)

(0, -6, 3)

Vector Spaces (Ch. 4)

100

Give an example of a vector space outside of R^n.

matrices, 

polynomials,

functions

200

In the vector space of continuous functions, give me a set of 3 functions that are linearly independent.

[sin x, cos x, x^2]

300

Give an example of a subspace of R^2.

Any line through the origin, y=mx.

400

4.4 (18)

p = 1(p_1) + 2(p_2) - (p_3)

500

1- Define rank and nullity. 

2- Given a matrix A, what's the relationship between rank, nullity, and the number of columns of A?

1- Rank is the number of leading 1's a matrix has when in RREF. Nullity is the dimension of the nullspace of a matrix. 

2- Rank + Nullity = # of columns

OR, # of leading 1's + free variables = # of columns

E.values/E.vectors (Ch. 5)

100

Define eigenvectors geometrically.

They are vectors whose direction doesn't change when multiplied by another matrix A.

200

Let theta, alpha, and sigma be the main diagonal elements in a lower triangular matrix. 

Find the eigenvalues of this matrix.

Theta, alpha, and sigma.

300

1- Given an nxn matrix, the most number of eigenvalues and eigenvectors it can have is ________________. 

2- Can a matrix have no eigenvalues?

1- n

2- No, a matrix will always have eigenvalues but it's not guaranteed that they will be real.

400

Find a 3x3 matrix whose eigenvalues are pi, e, and 2.5.

Any 3x3 triangular matrix with pi, e, and 2.5 along the main diagonal.

500

Find the eigenvalues in 5.1 (6b).

2

Inner Product Spaces (Ch. 6)

100

What's an inner product?

Another operation defined upon vector spaces where 2 vectors operate with each other and the output is a scalar.

200

6.1 (10)

56

300

6.1 (38a)

-4/7

400

6.1 (38c)

(8/7)^.5

500

6.1 (40)

1