Linear Algebra Test 2

Question 1

Elementary Row Operation

Answer 1

Given a matrix A the following row operations are called elementary:
a) swapping two rows,
b) multiplying a row by a non-zero scalar,
c) and adding a scalar multiple of a row to another row.

Question 2

Elementary Matrix

Answer 2

A matrix obtained by performing an elementary row operation on the identity matrix is called a _____.

Question 3

Upper Echelon Form

Answer 3

A matrix is said to be in _____ if
1) all zero rows are below all non-zero rows and
2) the first non-zero entry of each row is strictly to the right of the first non-zero entry of the row above

Question 4

Reduced Upper Echelon Form

Answer 4

A matrix is said to be in _____ if it is in upper echelon form and
1) the first non-zero entry of each row is 1 and
2) all entries in the same column as the first non-zero entry of a row are zero

Question 5

Determinant of a 2-by-2 Matrix

Answer 5

The _____ of a 2-by-2 matrix A = (a b c d) , denoted detA, is defined as detA = ad - bc.

Question 6

Fundamental Properties of 2-by-2 Determinants

Answer 6

Let F be a field. Viewed as a map det : F^2 x F^2 -> F, i.e. for u, v ∈ F^2 we write det(u|v) to mean that det(u1 v1 u2 v2), the determinant satisfies
1) bilinearity, i.e. det(|v) and det(u|) are linear for any fixed u, v ∈ F^2 and
2) anti-symmetry, i.e. det(u|v) = - det(v|u) for any u, v ∈ F^2

Question 7

Permutation

Answer 7

An invertible map σ : {1,…,n} -> {1,…,n} is called a _____. The set of all ____ of {1,…,n} is denoted by Sn

Question 8

Cycle

Answer 8

Let i1, . . . , ik be distinct elements of {1,…,n}. The k-cycle denoted by σ = (i1…ik) is the element σ of Sn defined by
- σ(ij ) = ij+1 if j < k and
- σ(ik) = i1

Question 9

Transposition

Answer 9

2 cycles are called _____.

Question 10

Sign of a Permutation

Answer 10

A permutation σ is called even/odd if it can be written as a composition of an even/odd number of transpositions. We define
sign σ = + 1 if σ is even and
− 1 if σ is odd

Question 11

Determinant of an n-by-n Matrix

Answer 11

The ____ A is defined to be detA = ∑σ∈Sn sign(σ)a1σ(1)a2σ(2) . . . anσ(n)

Question 12

Transpose of a Matrix

Answer 12

Let A be an n-by-m matrix. Its ____, denoted A^T, is the m-by-n matrix whose entries are A^Tij = Aji (i.e. columns and rows are swapped)

Question 13

Eigenvalue

Answer 13

Let T : V → V be a linear map. Let v be a nonzero vector in V and let λ be a scalar such that T (v) = λv. We call v an eigenvector of T and call λ and _____ of T

Question 13

Eigenvector

Answer 13

Let T : V → V be a linear map. Let v be a nonzero vector in V and let λ be a scalar such that T (v) = λv. We call v an _____ of T and call λ and eigenvalue of T

Question 14

Characteristic Polynomial

Answer 14

Let A be a square matrix. The polynomial pA(λ) = det(A − λI) is called the characteristic polynomial of A.

Question 15

Diagonal

Answer 15

A square matrix A is called
1. diagonal if aij = 0 when i ̸ = j and

Question 16

Diagonalisable

Answer 16

A square matrix A is called
2. diagonalisable if it is similar to a diagonal matrix