Linear Algebra: UNIT II: Linear Transformations and Diagonalization
Linear Transformation and Diagonalization: Multiple Choice Questions and Answers
Linear Algebra
Linear Algebra: UNIT II: Linear Transformations and Diagonalization : Multiple Choice Questions and Answers
LINEAR
TRANSFORMATION AND DIAGONALIZATION
Multiple Choice Questions and
Answers
1. Let V and W be
vector spaces over F. A function T: V→ W a linear transformation from V to W if
for all x, y ∈
V, c ∈
F, then
(A)
T(xy)=T(x)T (y)
(B)
T(x+y)=T(x) + T(y)
(C)
T(x/y) = T(x)/T(y)
(D)
T(v) = T (w)
2. Let V and W be two
vector spaces, let T: V→ W be linear, then the null space of T is such that
(A) N(T)
= {x ∈ V/T (x) = 0}
(B)
N (T) = { x, y ∈
V/F=0}
(C)
N(T) = {T(x)/x ∈
V}
(D)
N(T) = { x, y ∈
V/T(x) = x }
3. Let V and W be
vector spaces, and let T: V→ W be linear. Then T is 1−1 if and only if
(A)
R(T) = 1
(B)
R(T) = 2
(C) N(T)
= 0
(D)
N(T) = 1
4. Let T: P3
(R) → P2 (R). Define T [f(x)]
= f '(x), Let β and γ be the standard
bases for P3(R) and P2(R) respectively. Then the matrix
[T] is
[Ans:(B)]
5. The eigenvalues of 
∈
M2×2(R) are
(A) (3, −
1)
(B)
(−3, 1)
(C)
(1/3, 1/2)
(D)
(4, 1)
6. The eigenvalues of 
∈
M2×2(R) are
(A) (−3,
5)
(B)
(3,−5)
(C)
(2,−5)
(D)
(2,−3)
7. The eigenvalues of 
∈
M3×3(R) are
(A)
(2, 3, 4)
(B)
(1, 2, 3)
(C)
(3, 4, 4)
(D) (3,
3, 4)
8. Let T be a linear
operator on a finite dimensional vector space V and let λ be an eigenvalue of T
having multiplicity m, then
(A)
0<dim (E)<n
(B) 1≤
dim (Eλ) ≤m
(C)
1≤ dim (Eλ)<n
(D) 0<dim (Eλ) < 1
9. Let A ∈ Mm×n(F). The
characteristic polynomial of the matrix A is called
(A)
f(t)= |A−I|
(B) f(t)= |A−λIn|
(C)
f(t)=[A−λI]X
(D)
f(t)=[A−I]X
10. Let V and W be two
vector spaces, and let T: V→W be linear. If V is finite dimensional, then
(A)
Nullity (T)+ rank (T)= dim (V)
(B)
Nullity (T) ‒rank (T)= dim (W)
(C)
Nullity (U)+ rank (V)= dim (V)
(D)
rank (T) ‒ Nullity (T) = 0
Linear Algebra: UNIT II: Linear Transformations and Diagonalization : Tag: maths, mathematics : Linear Algebra - Linear Transformation and Diagonalization: Multiple Choice Questions and Answers
Linear Algebra: UNIT II: Linear Transformations and Diagonalization
Under Subject
Linear Algebra
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