Linear Algebra part B

Question 1

Invariant Subspace

Answer 1

Suppose T ∈ L(V).
A subspace U of V is invariant under T
if u ∈ U implies T(u) ∈ U

Question 2

Eigenvalue

Answer 2

Suppose T ∈ L(V).
A number lamda ∈ F is called an eigenvalue of T
if there exists v ∈ V
such that v ≠ 0 and Tv = lamda v

Question 3

Eigenvector

Answer 3

Suppose T ∈ L(V) and lamda ∈ F is an eigenvalue of T .
A vector v ∈ V is called an eigenvector of T
corresponding to lamda
if v ≠ 0 and Tv = lamda v

Question 4

Equivalent conditions to be an eigenvalue

Answer 4

Suppose V is finite dimensional
T ∈ L(V) and lamda ∈ F
Then the following are equivalent:
(a) lamda is an eigenvalue of T;
(b) T-lamda(I) is not injective;
(c) T-lamda(I) is not surjective;
(d) T-lamda(I) is not invertible;

Question 5

Linearly Independent Eigenvectors

Answer 5

Let T ∈ L(V)
Suppose lamda_(1),…,lamda_(m) are
distinct eigenvalues of T
and v_(1),…,v_(m) are corresponding eigenvectors.
Then v_(1),…,v_(m) is linearly independent

Question 6

Number of Eigenvalues

Answer 6

Suppose V is finite-dimensional
Then each operator on V has
at most dim V distinct eigenvalues

Ex- if dim V= 3 then each operator on V has at most 3 distinct eigenvalues

Question 7

Restriction Operator

Answer 7

T | _(U )
Suppose T ∈ L(V) and U is a subspace of V
invariant under T

The restrictive operator T | (U ) ∈ L(U) is defined by
T|(U)(u) = Tu

for u ∈ U

Question 8

Operator on complex vector spaces have an eigenvalue

Answer 8

Every operator on a finite-dimensional, nonzero, complex vector space has an eigenvalue

Question 9

Diagonal of a Matrix

Answer 9

The diagonal of a square matrix consists of the entries along the line from the upper left corner to the bottom right corner

Question 10

Upper Triangular Matrix

Answer 10

A matrix is called upper triangular if all the entries below the diagonal are equal to zero

Question 11

Determining of eigenvalues from upper triangular matrix

Answer 11

Suppose T ∈ L(V) has an upper triangular matrix with respect to some basis of V. Then the eigenvalues of T are precisely the entries on the diagonal of the upper triangular matrix

Question 12

Diagonal Matrix

Answer 12

A diagonal matrix is a square matrix that is 0 everywhere except possibly along the diagonal