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?

    <p>rank(A) = 3</p> Signup and view all the answers

    What is the condition for aij in the matrix A?

    <p>aij is an element of Z3</p> Signup and view all the answers

    What is the property of aij in the matrix W?

    <p>aij is 0 if i is even</p> Signup and view all the answers

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

    <p>n</p> Signup and view all the answers

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

    <p>n^2 - n</p> Signup and view all the answers

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

    <p>28</p> 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))?

    <p>3</p> Signup and view all the answers

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

    <p>24</p> 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?

    <p>4</p> Signup and view all the answers

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

    <p>{(a,b,c) | a,b,c rational}</p> Signup and view all the answers

    What is the dimension of the vector space W?

    <p>4</p> 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?

    <p>(3^4 – 1) (3^4 – 2) (3^4 – 3)</p> Signup and view all the answers

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

    <p>3</p> Signup and view all the answers

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

    <p>3</p> Signup and view all the answers

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

    <p>∞</p> Signup and view all the answers

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

    <p>∞</p> Signup and view all the answers

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

    <p>4</p> 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}?

    <p>4</p> Signup and view all the answers

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

    <p>2</p> 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?

    <p>3</p> 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?

    <p>n^2/2</p> Signup and view all the answers

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

    <p>One vector can be expressed as a linear combination of the others</p> 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?

    <p>40</p> Signup and view all the answers

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

    <p>3</p> Signup and view all the answers

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

    <p>The matrix has rank equal to the number of columns</p> Signup and view all the answers

    What is the vector space over F3?

    <p>F3 × F3</p> Signup and view all the answers

    What is the field of 3 elements?

    <p>F3</p> Signup and view all the answers

    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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    Linear Algebra DPP-15 PDF

    Description

    Practice problems in Linear Algebra, covering topics such as subspaces and matrix ranks. This quiz is part of the SAAKAAR 2.0 batch.

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