Linear Algebra, Numerical and Complex Analysis (MA11004)
Department of Mathematics
Indian Institute of Technology Kharagpur
Tutorial sheet 1, Spring 2024
Topics: Vector spaces over real and complex numbers; Subspaces; Linear combination; Span-
ning set; Linear dependence and independence of vectors.
1. Determine which of the following sets form vector spaces under the given operations:
(i) The set of all triples of real numbers (x, y, z) with the operations
(x, y, z) + (x0 , y 0 , z 0 ) = (x + x0 , y + y 0 , z + z 0 ) and k(x, y, z) = (kx, y, z)
for all k ∈ R and (x, y, z), (x0 , y 0 , z 0 ) ∈ R3 .
(ii) The set V = {(x1 , x2 ) ∈ R2 : x1 + x2 = 1, 0 ≤ x1 ≤ 1, 0 ≤ x2 ≤ 1} with the
operations
x1 + y1 x2 + y2
(x1 , x2 ) + (y1 , y2 ) = , and r(x1 , x2 ) = (rx1 , rx2 )
2 2
for all r ∈ R and (x1 , x2 ), (y1 , y2 ) ∈ V .
(iii) The set V of all positive real numbers x with the operations
x + x0 = xx0 and kx = xk for all k ∈ R and x, x0 ∈ V.
a 1
(iv) The set V of all 2×2 matrices of the form over R with usual matrix addition
1 b
and scalar multiplication.
(v) The set V = {f ∈ C(R) : ∃p ∈ N, f (x + p) = f (x), ∀x ∈ R}. Does V form a
vector space under the usual addition and scalar multiplication of C(R), the set of
all continuous functions over R?
2. Determine which of the following subsets are the subspaces of the given vector spaces:
(i) All vectors of the form (a, b, c), with b = a + c in R3 .
(ii) All matrices with A = AT in Mn×n (R), where Mn×n (R) is the vector space of all
n × n matrices over R.
a b
(iii) All matrices of the form with a + d = 0 in M2×2 (R).
c d
(iv) All matrices with det(A) = 0 in Mn×n (R).
(v) All vectors of the form (a, b, c), where ab = 0, in R3 .
(vi) Is the set W = {(a, b, c) : a3 = b3 } a subspace of both R3 and C3 ?
(vii) Is U a subspace of Mn×n (R) for a fix A ∈ Mn×n (R), where
U = {B ∈ Mn×n (R) : AB = BA}?
1
, R1
3. (a) For a fixed b ∈ R, let S = {f ∈ C[0, 1] : 0 f (x)dx = b}. Show that S is a subspace
of C[0, 1] if and only if b = 0.
(b) Let C[−4, 4] be the space of all continuous real-valued functions on [−4, 4]. Let
W ⊂ C[−4, 4] be the set of differentiable functions f on the interval (−4, 4) such
that f 0 (−1) = 3f (2). Show that W is a subspace of C[−4, 4].
4. Let V be a vector space over the field R of real numbers, and W1 , W2 be two subspaces
of V . Show that W1 ∩ W2 is a subspace of V . Give an example to show that W1 ∪ W2
need not be a subspace of V .
3 −1
5. (a) Write E = as a linear combination of
1 −2
1 1 1 1 1 −1
A= ,B = and C = .
0 −1 −1 0 0 0
(b) Write p = 2 + 2x + 3x2 as a linear combination of
p1 = 2 + x + 4x2 , p2 = 1 − x + 3x2 and p3 = 3 + 2x + 5x2 .
(c) Which of the following are linear combinations of the vectors u = (1, −1, 3) and
v = (2, 4, 0): (i) (3, 3, 3), (ii) (4, 2, 6), (iii) (1, 5, 6), (iv) (0, 0, 0).
6. In the vector space R3 , let u1 = (1, 2, 1), u2 = (3, 1, 5), u3 = (3, −4, 7). Then show that
span{u1 , u2 } = span{u1 , u2 , u3 }.
7. (a) Let S = {v1 , v2 , v3 , v4 } spans a vector space V . Show that the set {v1 − v2 , v2 −
v3 , v3 − v4 , v4 } also spans V .
(b) Let S = {u1 , u2 , u3 }, T = {u1 , u1 +u2 , u1 +u2 +u3 } and U = {u1 +u2 , u2 +u3 , u3 +u1 }
in R4 . Show that span S = span T = span U .
8. Determine which of the following sets are linear independent:
(a) {(4, −4, 8, 0), (2, 2, 4, 0), (6, 0, 0, 2), (6, 3, −3, 0)} in R4 .
(b) {2, 4 sin2 x, cos2 x} in C[−π, π].
(c) {t3 − 5t2 − 2t + 3, t3 − 4t2 − 3t + 4, 2t3 − 7t2 − 7t + 9} in P3 , where P3 is the set of
polynomials over R with degree at most 3.
9. Let f1 , f2 ∈ C[−1, 1] be defined as f1 (t) = t, t ∈ [−1, 1] and
−t if t ∈ [−1, 0],
f2 (t) =
t if t ∈ [0, 1].
Show that the set {f1 , f2 } is linearly dependent in C[0, 1] and in C[−1, 0], but linearly
independent in C[−1, 1].
10. Show that the set {1 + i, 1 − i} ⊂ C of vectors is linearly independent if C is taken as a
vector space over R. But it becomes linearly dependent when C is a vector space over C.
11. Show that u1 , u2 , . . . , uk ∈ Rn are linearly independent if and only if Au1 , Au2 , . . . , Auk
are linearly independent for any invertible n × n matrix A.
12. Let V be a vector space over a field F. Let A and B be two non-empty subsets of V.
Prove or disprove: Span(A) ∩ Span(B) 6= {0} =⇒ A ∩ B 6= φ, where Span(A) denotes
the spanning set of A.
——————————————-
2
, MA11004 - Linear Algebra, Numerical and Complex Analysis
Department of Mathematics
Indian Institute of Technology Kharagpur
Tutorial sheet 1, Spring 2024
1. (i) Check the distributive property of scalar-vector multiplication over scalar addition.
(ii) Check associative property.
(iii) Check the properties of vector space from the definition.
(iv) Check the closure property of vector addition.
(v) Check the usual properties of vector space for the two functions f1 , f2 with period
p1 , p2 and check the linearity property for f1 + f2 taking period p0 = lcm(p1 , p2 ).
2. (i) {(a, a + c, c) : a, c ∈ R} forms subspace in R3 .
(ii) (A + B)T = AT + B T and (αA)T = αAT .
a b
(iii) S = forms a subspace in M2×2 (R).
c −a
(iv) Give a counterexample where det(A1 + A2 ) 6= 0, but det(Ai ) = 0.
(v) (a1 , 0, c1 ), (0, b2 , c2 ) ∈ S, but their sum need not be there in S.
(vi) W is a subspace of R3 , but W is not a subspace of C3 .
(vii) U is a subspace of Mn×n (R).
3. (a) Every subspace contains the null vector (the additive identity) and for the converse
use the linearity of integration.
(b) Take g(x) = αf1 (x) + βf2 (x) and show that g 0 (−1) = 3g(2).
4. Verify the subspace criteria. What about W1 as x-axis and W2 as y-axis in R2 ?
5. (a) For E = αA + βB + γC, find α, β and γ.
(b) Take p = αp1 + βp2 + γp3 and find α, β and γ.
(c) Take each vector as αu + βv and solve for α and β.
6. Solve the expression u3 = αu1 + βu2 for some α, β ∈ R.
7. (a) Find the scalars α, β, γ, δ such that α(v1 − v2 ) + β(v2 − v3 ) + γ(v3 − v4 ) + δv4 =
av1 + bv2 + cv3 + dv4 holds.
(b) Find the scalars α, β, γ such that αu1 + β(u1 + u2 ) + γ(u1 + u2 + u3 ) = au1 + bu2 + cu3
holds.
8. (a) Linearly independent. Take c1 (4, −4, 8, 0)+c2 (2, 2, 4, 0)+c3 (6, 0, 0, 2)+c4 (6, 3, −3, 0) =
(0, 0, 0, 0) and c1 = c2 = c3 = c4 = 0 is the only solution.
(b) Linearly dependent. Consider c1 ·2+c2 ·4 sin2 x+c3 ·cos2 x = 0, and then differentiate
it with respect to x two times and find the values of ci from these three equations.
(c) Linearly dependent. Follow the same steps as (b).
9. In each of the intervals [−1, 0] and [0, 1], one has that f1 = cf2 for some scalar c, but in
the interval [−1, 1], do you get the same?
1
Department of Mathematics
Indian Institute of Technology Kharagpur
Tutorial sheet 1, Spring 2024
Topics: Vector spaces over real and complex numbers; Subspaces; Linear combination; Span-
ning set; Linear dependence and independence of vectors.
1. Determine which of the following sets form vector spaces under the given operations:
(i) The set of all triples of real numbers (x, y, z) with the operations
(x, y, z) + (x0 , y 0 , z 0 ) = (x + x0 , y + y 0 , z + z 0 ) and k(x, y, z) = (kx, y, z)
for all k ∈ R and (x, y, z), (x0 , y 0 , z 0 ) ∈ R3 .
(ii) The set V = {(x1 , x2 ) ∈ R2 : x1 + x2 = 1, 0 ≤ x1 ≤ 1, 0 ≤ x2 ≤ 1} with the
operations
x1 + y1 x2 + y2
(x1 , x2 ) + (y1 , y2 ) = , and r(x1 , x2 ) = (rx1 , rx2 )
2 2
for all r ∈ R and (x1 , x2 ), (y1 , y2 ) ∈ V .
(iii) The set V of all positive real numbers x with the operations
x + x0 = xx0 and kx = xk for all k ∈ R and x, x0 ∈ V.
a 1
(iv) The set V of all 2×2 matrices of the form over R with usual matrix addition
1 b
and scalar multiplication.
(v) The set V = {f ∈ C(R) : ∃p ∈ N, f (x + p) = f (x), ∀x ∈ R}. Does V form a
vector space under the usual addition and scalar multiplication of C(R), the set of
all continuous functions over R?
2. Determine which of the following subsets are the subspaces of the given vector spaces:
(i) All vectors of the form (a, b, c), with b = a + c in R3 .
(ii) All matrices with A = AT in Mn×n (R), where Mn×n (R) is the vector space of all
n × n matrices over R.
a b
(iii) All matrices of the form with a + d = 0 in M2×2 (R).
c d
(iv) All matrices with det(A) = 0 in Mn×n (R).
(v) All vectors of the form (a, b, c), where ab = 0, in R3 .
(vi) Is the set W = {(a, b, c) : a3 = b3 } a subspace of both R3 and C3 ?
(vii) Is U a subspace of Mn×n (R) for a fix A ∈ Mn×n (R), where
U = {B ∈ Mn×n (R) : AB = BA}?
1
, R1
3. (a) For a fixed b ∈ R, let S = {f ∈ C[0, 1] : 0 f (x)dx = b}. Show that S is a subspace
of C[0, 1] if and only if b = 0.
(b) Let C[−4, 4] be the space of all continuous real-valued functions on [−4, 4]. Let
W ⊂ C[−4, 4] be the set of differentiable functions f on the interval (−4, 4) such
that f 0 (−1) = 3f (2). Show that W is a subspace of C[−4, 4].
4. Let V be a vector space over the field R of real numbers, and W1 , W2 be two subspaces
of V . Show that W1 ∩ W2 is a subspace of V . Give an example to show that W1 ∪ W2
need not be a subspace of V .
3 −1
5. (a) Write E = as a linear combination of
1 −2
1 1 1 1 1 −1
A= ,B = and C = .
0 −1 −1 0 0 0
(b) Write p = 2 + 2x + 3x2 as a linear combination of
p1 = 2 + x + 4x2 , p2 = 1 − x + 3x2 and p3 = 3 + 2x + 5x2 .
(c) Which of the following are linear combinations of the vectors u = (1, −1, 3) and
v = (2, 4, 0): (i) (3, 3, 3), (ii) (4, 2, 6), (iii) (1, 5, 6), (iv) (0, 0, 0).
6. In the vector space R3 , let u1 = (1, 2, 1), u2 = (3, 1, 5), u3 = (3, −4, 7). Then show that
span{u1 , u2 } = span{u1 , u2 , u3 }.
7. (a) Let S = {v1 , v2 , v3 , v4 } spans a vector space V . Show that the set {v1 − v2 , v2 −
v3 , v3 − v4 , v4 } also spans V .
(b) Let S = {u1 , u2 , u3 }, T = {u1 , u1 +u2 , u1 +u2 +u3 } and U = {u1 +u2 , u2 +u3 , u3 +u1 }
in R4 . Show that span S = span T = span U .
8. Determine which of the following sets are linear independent:
(a) {(4, −4, 8, 0), (2, 2, 4, 0), (6, 0, 0, 2), (6, 3, −3, 0)} in R4 .
(b) {2, 4 sin2 x, cos2 x} in C[−π, π].
(c) {t3 − 5t2 − 2t + 3, t3 − 4t2 − 3t + 4, 2t3 − 7t2 − 7t + 9} in P3 , where P3 is the set of
polynomials over R with degree at most 3.
9. Let f1 , f2 ∈ C[−1, 1] be defined as f1 (t) = t, t ∈ [−1, 1] and
−t if t ∈ [−1, 0],
f2 (t) =
t if t ∈ [0, 1].
Show that the set {f1 , f2 } is linearly dependent in C[0, 1] and in C[−1, 0], but linearly
independent in C[−1, 1].
10. Show that the set {1 + i, 1 − i} ⊂ C of vectors is linearly independent if C is taken as a
vector space over R. But it becomes linearly dependent when C is a vector space over C.
11. Show that u1 , u2 , . . . , uk ∈ Rn are linearly independent if and only if Au1 , Au2 , . . . , Auk
are linearly independent for any invertible n × n matrix A.
12. Let V be a vector space over a field F. Let A and B be two non-empty subsets of V.
Prove or disprove: Span(A) ∩ Span(B) 6= {0} =⇒ A ∩ B 6= φ, where Span(A) denotes
the spanning set of A.
——————————————-
2
, MA11004 - Linear Algebra, Numerical and Complex Analysis
Department of Mathematics
Indian Institute of Technology Kharagpur
Tutorial sheet 1, Spring 2024
1. (i) Check the distributive property of scalar-vector multiplication over scalar addition.
(ii) Check associative property.
(iii) Check the properties of vector space from the definition.
(iv) Check the closure property of vector addition.
(v) Check the usual properties of vector space for the two functions f1 , f2 with period
p1 , p2 and check the linearity property for f1 + f2 taking period p0 = lcm(p1 , p2 ).
2. (i) {(a, a + c, c) : a, c ∈ R} forms subspace in R3 .
(ii) (A + B)T = AT + B T and (αA)T = αAT .
a b
(iii) S = forms a subspace in M2×2 (R).
c −a
(iv) Give a counterexample where det(A1 + A2 ) 6= 0, but det(Ai ) = 0.
(v) (a1 , 0, c1 ), (0, b2 , c2 ) ∈ S, but their sum need not be there in S.
(vi) W is a subspace of R3 , but W is not a subspace of C3 .
(vii) U is a subspace of Mn×n (R).
3. (a) Every subspace contains the null vector (the additive identity) and for the converse
use the linearity of integration.
(b) Take g(x) = αf1 (x) + βf2 (x) and show that g 0 (−1) = 3g(2).
4. Verify the subspace criteria. What about W1 as x-axis and W2 as y-axis in R2 ?
5. (a) For E = αA + βB + γC, find α, β and γ.
(b) Take p = αp1 + βp2 + γp3 and find α, β and γ.
(c) Take each vector as αu + βv and solve for α and β.
6. Solve the expression u3 = αu1 + βu2 for some α, β ∈ R.
7. (a) Find the scalars α, β, γ, δ such that α(v1 − v2 ) + β(v2 − v3 ) + γ(v3 − v4 ) + δv4 =
av1 + bv2 + cv3 + dv4 holds.
(b) Find the scalars α, β, γ such that αu1 + β(u1 + u2 ) + γ(u1 + u2 + u3 ) = au1 + bu2 + cu3
holds.
8. (a) Linearly independent. Take c1 (4, −4, 8, 0)+c2 (2, 2, 4, 0)+c3 (6, 0, 0, 2)+c4 (6, 3, −3, 0) =
(0, 0, 0, 0) and c1 = c2 = c3 = c4 = 0 is the only solution.
(b) Linearly dependent. Consider c1 ·2+c2 ·4 sin2 x+c3 ·cos2 x = 0, and then differentiate
it with respect to x two times and find the values of ci from these three equations.
(c) Linearly dependent. Follow the same steps as (b).
9. In each of the intervals [−1, 0] and [0, 1], one has that f1 = cf2 for some scalar c, but in
the interval [−1, 1], do you get the same?
1
