Linear Algebra DPP-15

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Questions and Answers

What is the order of the matrix A?

4x3 (correct)
10x10
3x4
5x5

What is the value of n in the expression n2 - n = 6?

4
2
3 (correct)
5

What is the dimension of the set W?

5x10
10x5 (correct)
10x10
5x5

What is the condition for a matrix A to be in the set X?

rank(A) = 3 (C) Signup and view all the answers

What is the condition for aij in the matrix A?

aij is an element of Z3 (A) Signup and view all the answers

What is the property of aij in the matrix W?

aij is 0 if i is even (D) Signup and view all the answers

What is the dimension of the solution space of a nth order linear homogeneous differential equation?

n (D) Signup and view all the answers

What is the dimension of the vector space of skew symmetric matrices of n × n of real entries?

n^2 - n (D) Signup and view all the answers

What is the number of subsets of 𝐹3 × 𝐹3 − {(0,0)} having exactly two elements?

28 (A) Signup and view all the answers

What is the value of p in the equation ∴ No.of L.I.subset = (pn – 1) (pn – p) (pn – p2)…..(pn – p(k-1))?

3 (B) Signup and view all the answers

What is the number of linearly independent subsets of order 2 in 𝐹3 × 𝐹3 − {(0,0)}?

24 (C) Signup and view all the answers

What is the number of distinct linearly dependent sets of the form {u, v}, where u, v ∈ F3 × F3 / {(0,0)} and u ≠ v?

4 (A) Signup and view all the answers

Which of the following sets is not a subspace of R3?

{(a,b,c) | a,b,c rational} (D) Signup and view all the answers

What is the dimension of the vector space W?

4 (B) Signup and view all the answers

What is the number of matrices of rank three in the space of all 4×3 matrices with entries in the finite field of three elements?

(3^4 – 1) (3^4 – 2) (3^4 – 3) (D) Signup and view all the answers

Let A be a matrix with rank 3. What is the number of linearly independent columns of A?

3 (B) Signup and view all the answers

What is the value of k in the equation subset S ⊆ V, |S| = k?

3 (D) Signup and view all the answers

What is the dimension of the vector space V = C5 [a,b] over R?

∞ (A) Signup and view all the answers

What is the dimension of the vector space C5 [a, b] over R?

∞ (A) Signup and view all the answers

What is the number of linearly dependent subsets of the form {𝑢, 𝑣} in 𝐹3 × 𝐹3 − {(0,0)}?

4 (A) Signup and view all the answers

What is the dimension of the vector space W = {f ∈ V : 4f + 2(df/dt) - (d^2f/dt^2) = 0}?

4 (C) Signup and view all the answers

What is the dimension of the subspace W, where W = {f ∈ V : 4f + 2f′ - f″ = 0}?

2 (D) Signup and view all the answers

Let A be a matrix with rank 3. What is the maximum number of linearly independent rows of A?

3 (C) Signup and view all the answers

What is the dimension of the vector space V = {A = [aij] (n×n) | aij ∈ C, aij = -aji} over R?

n^2/2 (B) Signup and view all the answers

What is the condition for a set of vectors to be linearly dependent?

One vector can be expressed as a linear combination of the others (C) Signup and view all the answers

What is the dimension of the subspace W = {[aij]: aij = 0 if i is even} of all 10×10 real matrices?

40 (D) Signup and view all the answers

What is the dimension of the vector space F3 × F3 / {(0,0)}?

3 (C) Signup and view all the answers

What is the condition for a matrix to have a unique solution?

The matrix has rank equal to the number of columns (A) Signup and view all the answers

What is the vector space over F3?

F3 × F3 (B) Signup and view all the answers

What is the field of 3 elements?

F3 (D) Signup and view all the answers

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Study Notes

Linear Algebra

Subspaces of R3

{(a, b, c) | a, b, c rational} is not a subspace of R3 because scalar multiplication fails.
{(0, 0, 0)} is a subspace of R3.
{(a, a + b, –a + 2b) | a, b real} and {(a, a – b, b) | a, b real} are subspaces of R3.

Vector Spaces

The vector space C5 [a, b] over R is the set of all functions f ∈ C [a, b] where the 5th derivative exists and is continuous.
The dimension of C5 [a, b] is infinite (∞).

Dimension of Vector Spaces

The dimension of the vector space V = {A = aij | aij ∈ C, aij = –aji} over the field R is n2 – n.
The dimension of the vector space W = {[aij] : aij = 0 if i is even} of all 10×10 real matrices is 50.

Linear Independence

The number of distinct linearly dependent sets of the form {u, v}, where u, v ∈ F3 × F3 / {(0,0)} and u ≠ v is 4.
The number of linearly independent subsets of order 2 in F3 × F3 is 24.

Matrix Rank and Linear Independence

The number of matrices of rank three in the space M of all 4×3 matrices with entries in the finite field of three elements is (34 – 1)(34 – 3)(34 – 32).

Skew Symmetric Matrices

The dimension of the vector space of skew symmetric matrices of n × n of real entries is n2 – n / 2.

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