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What is the order of the matrix A?
What is the order of the matrix A?
What is the value of n in the expression n2 - n = 6?
What is the value of n in the expression n2 - n = 6?
What is the dimension of the set W?
What is the dimension of the set W?
What is the condition for a matrix A to be in the set X?
What is the condition for a matrix A to be in the set X?
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What is the condition for aij in the matrix A?
What is the condition for aij in the matrix A?
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What is the property of aij in the matrix W?
What is the property of aij in the matrix W?
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What is the dimension of the solution space of a nth order linear homogeneous differential equation?
What is the dimension of the solution space of a nth order linear homogeneous differential equation?
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What is the dimension of the vector space of skew symmetric matrices of n × n of real entries?
What is the dimension of the vector space of skew symmetric matrices of n × n of real entries?
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What is the number of subsets of 𝐹3 × 𝐹3 − {(0,0)} having exactly two elements?
What is the number of subsets of 𝐹3 × 𝐹3 − {(0,0)} having exactly two elements?
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What is the value of p in the equation ∴ No.of L.I.subset = (pn – 1) (pn – p) (pn – p2)…..(pn – p(k-1))?
What is the value of p in the equation ∴ No.of L.I.subset = (pn – 1) (pn – p) (pn – p2)…..(pn – p(k-1))?
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What is the number of linearly independent subsets of order 2 in 𝐹3 × 𝐹3 − {(0,0)}?
What is the number of linearly independent subsets of order 2 in 𝐹3 × 𝐹3 − {(0,0)}?
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What is the number of distinct linearly dependent sets of the form {u, v}, where u, v ∈ F3 × F3 / {(0,0)} and u ≠ v?
What is the number of distinct linearly dependent sets of the form {u, v}, where u, v ∈ F3 × F3 / {(0,0)} and u ≠ v?
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Which of the following sets is not a subspace of R3?
Which of the following sets is not a subspace of R3?
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What is the dimension of the vector space W?
What is the dimension of the vector space W?
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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?
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?
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Let A be a matrix with rank 3. What is the number of linearly independent columns of A?
Let A be a matrix with rank 3. What is the number of linearly independent columns of A?
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What is the value of k in the equation subset S ⊆ V, |S| = k?
What is the value of k in the equation subset S ⊆ V, |S| = k?
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What is the dimension of the vector space V = C5 [a,b] over R?
What is the dimension of the vector space V = C5 [a,b] over R?
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What is the dimension of the vector space C5 [a, b] over R?
What is the dimension of the vector space C5 [a, b] over R?
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What is the number of linearly dependent subsets of the form {𝑢, 𝑣} in 𝐹3 × 𝐹3 − {(0,0)}?
What is the number of linearly dependent subsets of the form {𝑢, 𝑣} in 𝐹3 × 𝐹3 − {(0,0)}?
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What is the dimension of the vector space W = {f ∈ V : 4f + 2(df/dt) - (d^2f/dt^2) = 0}?
What is the dimension of the vector space W = {f ∈ V : 4f + 2(df/dt) - (d^2f/dt^2) = 0}?
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What is the dimension of the subspace W, where W = {f ∈ V : 4f + 2f′ - f″ = 0}?
What is the dimension of the subspace W, where W = {f ∈ V : 4f + 2f′ - f″ = 0}?
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Let A be a matrix with rank 3. What is the maximum number of linearly independent rows of A?
Let A be a matrix with rank 3. What is the maximum number of linearly independent rows of A?
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What is the dimension of the vector space V = {A = [aij] (n×n) | aij ∈ C, aij = -aji} over R?
What is the dimension of the vector space V = {A = [aij] (n×n) | aij ∈ C, aij = -aji} over R?
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What is the condition for a set of vectors to be linearly dependent?
What is the condition for a set of vectors to be linearly dependent?
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What is the dimension of the subspace W = {[aij]: aij = 0 if i is even} of all 10×10 real matrices?
What is the dimension of the subspace W = {[aij]: aij = 0 if i is even} of all 10×10 real matrices?
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What is the dimension of the vector space F3 × F3 / {(0,0)}?
What is the dimension of the vector space F3 × F3 / {(0,0)}?
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What is the condition for a matrix to have a unique solution?
What is the condition for a matrix to have a unique solution?
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What is the vector space over F3?
What is the vector space over F3?
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What is the field of 3 elements?
What is the field of 3 elements?
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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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Description
Practice problems in Linear Algebra, covering topics such as subspaces and matrix ranks. This quiz is part of the SAAKAAR 2.0 batch.
