Tag : linear algebra

Let V be a nonzero subspace of the complex vector space M_7(C) such that every nonzero matrix in V is invertible. Then, the dimension of V over C is
Let a = [(1/sqrt{3}), (-1/sqrt{2}), (1/sqrt{6}), (0)]. Consider the following two statements. P: The matrix I_4 - aa^T is invertible. Q: The matrix I_4 - 2aa^T is invertible. Then, which one of the following holds?
Let A be a 6 × 5 matrix with entries in R and B be a 5 × 4 matrix with entries in R. Consider the following two statements. P: For all such nonzero matrices A and B, there is a nonzero matrix Z such that AZB is the 6 × 4 zero matrix. Q: For all such nonzero matrices A and B, there is a nonzero matrix Y such that BYA is the 5 × 5 zero matrix. Which one of the following holds?
Let F_{11}(x) be the real vector space of polynomials, in the variable x with real coefficients and having degree at most 11, together with the zero polynomial. Let E = {s_{0}(x), s_{1}(x), . . . , s_{11}(x)}, F = {r_{0}(x), r_{1}(x), . . . , r_{11}(x)} be subsets of P_{11}(x) having 12 elements each and satisfying, s_{0}(3) = s_{1}(3), = . . . = s_{11}(3) = 0, r_{0}(4) = r_{1}(4) = . . . = r_{11}(4) = 1. Then, which one of the following is TRUE?
Let 𝑃_7(𝑥) be the real vector space of polynomials, in the variable 𝑥 with real coefficients and having degree at most 7, together with the zero polynomial. Let 𝑇: 𝑃_7(𝑥) → 𝑃_7(𝑥) be the linear transformation defined by T(f(x)) = f(x) + df(x)/dx. Then, which one of the following is TRUE?
For a matrix 𝑀, let Rowspace(𝑀) denote the linear span of the rows of 𝑀 and Colspace(𝑀) denote the linear span of the columns of 𝑀. Which of the following hold(s) for all 𝐴, 𝐵, 𝐶 ∈ 𝑀_{10}(ℝ) satisfying 𝐴 = 𝐵𝐶 ?
Consider the 4 × 4 matrix M = ([0, 1, 2, 3], [1, 0, 1, 2], [2, 1, 0, 1], [3, 2, 1, 0]). If a_{i,j} denotes the (i,j)^{th} entry of M^{-1}, then a_{4,1} equals _____________ (rounded off to two decimal places).
Let 𝑃_{12}(𝑥) be the real vector space of polynomials in the variable 𝑥 with real coefficients and having degree at most 12, together with the zero polynomial. Define V = {f ∈ P_{12}(x): f(-x) = f(x) for all x ∈ R and f(2024) = 0}. Then, the dimension of V is _____________
Let M = ([0, 0, 0, 0, -1], [2, 0, 0, 0, -4], [0, 2, 0, 0, 0], [0, 0, 2, 0, 3], [0, 0, 0, 2, 2]). If 𝑝(𝑥) is the characteristic polynomial of 𝑀, then 𝑝(2) − 1 equals _____________
Consider the following statements. P : If a system of linear equations Ax = b has a unique solution, where A is an m×n matrix and b is an m × 1 matrix, then m = n. Q : For a subspace W of a nonzero vector space V, whenever u ∈ V \ W and v ∈ V \ W, then u + v ∈ V\W. Which one of the following holds?