definitions sem2-half2

Question 1

what is R^n

Answer 1

is called a n-dimensional vector space. It is the set of n-dimensional vectors
i.e. the set of all n × 1-matrices
R^n =
{
[r1
r2
…
rn]
| r1,r2,rn e R}

Question 2

Span{v}

Answer 2

Let v be a n-dimensional vector. The span of v is the following set of vectors span {v} = {λ · v | λ ∈ R} (the space spanned by v)

Question 3

Span{v, w}

Answer 3

Let v, w be two n-dimensional vectors. The span of v and w is the set
span {v, w} = {λ1 · v + λ2 · w | λ1, λ2 ∈ R} (the space spanned by v and w)

Question 4

Span{v1, v2, . . . , vm}

Answer 4

Let v1, v2, . . . , vm be some n-dimensional vectors. The span of
v1, v2, . . . , vm is the following set of vectors
span {v1, v2, . . . , vm} =
{
λ1 ·v1 + λ2 ·v2 + · · · + λm ·vm | λ1, λ2, . . . , λm ∈ R
}

Question 5

Basis of R^2

Question 6

What does it mean when we say “ λ is an eigenvalue of A

Answer 6

A real number λ is an eigenvalue of A if there is a non-zero vector [
x1
x2
]
such that
A * [X1 X2] = λ [X1 X2]

Question 7

eigenvector of A

Answer 7

Any vector [x1 x2] is an eigenvector of A corresponding to λ if
1- non-zero vector
2- eigen form

Question 8

A is a diagonal matrix

Answer 8

An n × n-matrix A is called a diagonal matrix, if the entries outside the main diagonal
are all zero, i.e.

Question 9

A is an invertible matrix

Answer 9

An n × n-matrix A is called an invertible matrix, if there exists an n × n-matrix A ^−1
such that
A · A^−1= diagonal matrix
A^−1 · A =diagonal matrix

Question 10

The matrix A is diagonalizable

Answer 10

An n × n matrix A is diagonalizable if there exists an invertible n × n matrix P such
that
P ^ −1 · A · P =diagonal matrix
P ^ −1 · A · P=(n × n-diagonal matrix.)

Question 11

P diagonalizes A

Answer 11

A matrix P diagonalizes A, if
* P is invertible and
*P ^ −1 · A · P=(n × n-diagonal matrix.)

Question 12

“f is a linear map

Answer 12

A function f : R^3 → R^ 37 is called linear map if for any two 3-dimensional vectors
v= [a1 a2 a3]
u = [b1 b2 b3]
where a1,a2,a3,b1,b2,b3 e R
and for any real number r the following two conditions are satisfied:
- f([v]+[u]) = f([v]) +f([u])
- f(r*[v]) = r * f([v])