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Download Sample Problems for Midterm 2 1 True or False: 1.1 If V is a vector
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Sample Problems for Midterm 2 1 True or False: 1.2 If V is a vector space, then (−1) u = −u for any vector u in V. The set of all matrices c db where ≥ 0 is a subspace of M2×2 . 1.3 If S = { v1 , v2 , v3 } is a spanning set, then T = { v1 , v3 } is also a spanning set. 1.4 If v 6= 0, then the set {v} is linearly independent. 1.5 If v3 is a linear combination of v1 and v2 , then spn{v1 , v2 , v3 } = spn{v1 , v2 }. 1.6 Every linearly independent set can be extended to a basis. 1.7 Every spanning set can be extended to a basis. 1.8 If {v1 , . . . , vn } is a spanning set and {w1 , . . . , wm } is linearly independent, then n ≤ m. 1.9 The matrices [1, 2, 3], [1, 0, 1], [1, 2, 3] are linearly independent. 1.1 1.10 The polynomials t, t 2 , t 3 span P3 . 1.11 A minimal spanning set is a basis. 1.12 A minimal independent set is a basis. 1.13 If dim V = n, then any independent set in V has at least n elements. 1.14 If dim V = n, then any independent set in V with n elements is a basis for V. 1.15 If W is a subspace of V, then dim W ≤ dim V. 1.16 If W = spn S and S is linearly independent, then S is a basis for W. 1.17 [v − w] S = [v] S − [w] S , for all vectors v, w, and all bases S. 1.18 Any transition matrix PS←T is non-singular. 1.19 Given three bases S1 , S2 , and S3 , we have that PS2 ←S1 · PS3 ←S2 = PS3 ←S1 . 1.20 If two matrices are row-equivalent, they they have the same row space. 1.21 If two matrices are column-equivalent, they they have the same row rank. 1.22 A linear system Ax = b, where A is n × n, has a unique solution if rnk A = n. 1.23 If rnk(A) = 0, then A is singular. 1.24 If nllity(A) = 0, then A is singular. 1.25 If an m × n matrix has rank n, then its columns are independent. 1.26 An n × n matrix has rank n if and only if its columns are a basis. 2 Give a proof of the following statements: 2.1 The set of all 1 × 3 matrices of the form [, b, − 2b] is a subspace of R3 . 2.2 2.3 The set of polynomials of the form t 4 − t is a subspace of P5 . The set of 2-vectors of the form 1 is not a subspace of R2 . 2.4 Let D be the disk in R2 centered at the origin with radius 1. Then D is not a subspace of R2 . 3 Consider the matrix: 1 1 4 1 2 0 A= 0 1 1 2 1 0 0 1 −1 0 0 1 6 0 1 2 2 1 2 3.1 Compute a basis for the row space of A. 3.2 Compute a basis for the column space of A. 3.3 Compute a basis for the kernel of A. 3.4 Compute rnk(A). 3.5 Compute nllity(A). 3.6 Is A non-singular? 3.7 How many solutions does Ax = b have? 4 Consider the following collections of polynomials: S={ t3 , t 2 + t, 2t 2 + 1, t+1 T={ } t3 + t2 , t, t3 1, 4.1 Show that both S and T are bases for P3 . 4.2 Find the transition matrix PS←T from the T-basis to the S-basis. 4.3 Find the transition matrix PT←S from the S-basis to the T-basis. 4.4 Let p(t) = t 3 + t 2 + t + 1. Compute the coordinate vector [p(t)] T . 4.5 Using your answers to previous questions, compute the coordinate vector [p(t)] S . 5 Let W be the subspace of P3 consisting of all polynomials of the form t 3 + b t 2 + c t + d, where 3 − b − 5d = 0 and 4 − c + d = 0. 5.1 Compute the dimension of W. 5.2 Compute a basis for W. 6 Consider the following set of vectors in R3 : S= ¦ 1 0 , 1 0 2 3 , 1 2 3 , 3 6.1 Show that S is linearly dependent. 6.2 Express one of the vectors in S as a linear combination of the others. 6.3 Find a basis for spn(S). 6.4 Find a subset of S that is a basis for spn(S). 7 S= 1 0 0 −1 , 0 0 1 0 , 2 0 1 −2 7.2 Show that S does not span M2×2 . Determine whether the matrix 11 11 belongs to spn(S). 7.3 Find a basis for spn(S). 7.4 Find a subset of S that is a basis for spn(S). 8 6 6 « Consider the following set of matrices in M2×2 . ¨ 7.1 } , 0 1 0 0 t+1 Consider the following set of polynomials in P3 : S= t 3 + t 2 + t + 1, t 2 + t + 1, 8.1 Show that S does not span P3 . 8.2 Find a polynomial in P3 that does not belong to Spn(S). 8.3 Find a basis for P3 that contains S. ©