ma1513 2

Question 1

p is a projection of vector v onto the plane

Answer 1

v - p is orthogonal to the plane

Question 2

projection p of a vector u onto V where V is a subspace of R^n</sub>

Answer 2

u-p is orthogonal to V i.e. dot product of (u-p) with every vector in V is 0

Question 3

T(0)

linear transformation

Answer 3

0

Question 4

T(u + v)

Answer 4

T(u) + T(v)

Question 5

T(cu)

Answer 5

cT(u)

Question 6

If x is an eigenvector associated with the eigenvalue λ, the eigenvalue of kx will be

Answer 6

λ

Question 7

Zero vector can be an eigenvector T/F

Answer 7

F

Question 8

0 can be an eigenvalue T/F

Answer 8

T

Question 9

Eigenvalues of a triangular matrix

Answer 9

All diagonal entries

Question 10

Eigenspace

Answer 10

Solution space of the homogeneous system (λI - A)x = 0

Question 11

Finding the basis for eigenspace

Answer 11

1. Solve the system (λI - A)x = 0 using GE
2. Get the general soln and separate the parameters
3. Write the associated vectors in the span

Question 12

Eigenvalue of I₃

Answer 12

Only one eigenvalue 1

Question 13

Eigenspace of I₃

Answer 13

R³

Question 14

If A is an n x n matrix, sum of multiplicities of eigenvalues

Answer 14

n

Question 15

relation bw dimension of eigenspace and multiplicity

Answer 15

dim E_λi <= r_i

Question 16

Condition for n x n matrix A to be diagonalizable

Answer 16

A has n linearly independent eigenvectors

Question 17

BD
where D is a diagonal matrix

Answer 17

BD = (b₁d₁ b₂d₂ … b_nd_n)

Question 18

A is an n x n matrix
|S| < n
where S is the total number of basis vectors from all eigenspaces

Answer 18

A is not diagonalizable

Question 19

A is an n x n matrix
|S| = n
where S is the total number of basis vectors from all eigenspaces

Answer 19

A is diagonalizable

Question 20

Algorithm for diagonalization

Answer 20

1. Solve the characteristic polynomial det(𝜆𝜆I – A) to find all distinct eigenvalues 𝜆₁, 𝜆₂, …, 𝜆_k.
2. For each 𝜆_i, find a basis S_𝜆𝑖 for the eigenspace E_𝜆𝑖 by solving (𝜆_iI – A)x = 0
3. Let S = S_𝜆1 ∪ S_𝜆2 ∪ … ∪ S_𝜆k . (Then |S| is the total number of basis vectors from all the
   eigespaces)

Question 21

relation bw dimension of eigenspace and multiplicities

Answer 21

dim E_λi = |S_λi|

dim E_λi <= r_i

Question 22

Condition when A is diagonalizable

in terms of E_λi and multiplicity

Answer 22

dim E_λi = r_i

then |S| = r₁ + r₂ + .. + r_k = n

Question 23

Condition when A is not diagonalizable

in terms of E_λi and multiplicity

Answer 23

dim E_λi < r_i

then |S| < r₁ + r₂ + .. + r_k = n

Question 24

remember this

Answer 24

when we carry out the diagonalization algorithm, after step 1, if we obtain n distinct eigenvalues, then we can straight away conclude that the matrix is
diagonalizable without going through the remaining steps

Question 25

remember this pt. 2

Answer 25

All diagonal matrices are automatically diagonalizable

Question 26

If A is a diagonalizable matrix with P^–1AP = D, then A^m =

Answer 26

PD^mP^–1