1.2: Which of the following is true?a) If S is a set of k vectors in " and kn, then S is linearly dependentb) IfS is a set of k vectors in R and k = n, then S is linearly independentc) Every subset of a set of linearly dependent vectors is linearly dependentd) If the column vectors of an n x n matrix A are linearly independent, then throw vectors of A are also linearly independentNone of the above

Which of the following option is true :-

Answered: 1.2: Which of the following is true? a)…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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5. Which of the following se:ntences is false a) The determinant «.f A-B equals det A – det B b) If A is not invertil.le then AB is not invertible c) The determinant of A is ± the product of its pivots d) AB and BA have the same determinant e) All sentences are false O O
4. a) Give an example of a set of nonzero vectors in R3 that is linearly independent but is not a basis for R3. Explain why your answer works.
4] Show that the following sets of vectors are linearly dependent in R3. (c) {{5, 2, -3), (3, 0, 4), (-3, 0, -4)} (d) {{1, 1, 1), (2, 2, 2), (0, 1, 5)}
Let A be an n x n matrix. If vi, 2, ..., Un are vectors in R" such that {Avi, Au,..., Av} is linearly independent, show that A is invertible.
B = {bj, ..., b, } be a basis for a vector space V. Explain why the B-coordinate vectors of b1, ..., b, are the columns e1, en of the n x n identity matrix.
If {v1, v2, . . . , vn} is a linearly dependent set of vectors, prove thatone of these vectors is a linear combination of the other.
If v1, ... , Vn are linearly independent, the space they span has dimension __ . These vectors are a __ for that space. If the vectors are the columns of an m by n matrix, then m is __ than n. If m = n, that matrix is __ .
TRUE FALSE Q 1)If V is a vector space with dimension n>0, then any n vectors in V spans V. 2)In a matrix A, the dimension of the row space of A is equal to the dimension of the column space of A. 3)An n×n matrix A is nonsingular if and only if the reduced row echelon form of A is the identity matrix I.
16. Show that if the non-zero vectors s), s2),..., s) are conjugate to a positive definite matrix G, then the vectors are independent.
Suppose now we have a set of non standard basis vectors v1, v2, .., Un e R" b) which satisfies the following equation: 1 1 WV = I = [v1, v2, ..., vn] where w and vi are the rows and columns of W and V, respectively. Any vector z € R" can be written as a linear combination of the st andard basis vectors v1, v2, ..., Vn as follows: k=1 Determine the coefficients ak in terms of z and wf only.
Prove that the nonzero row vectors of a matrix in row-echelon form are linearly independent. Let = A be an m x n matrix in row-echelon form. If the first column of A is not all zero and e,,, e,m, ean denote leading ones, then the nonzero row vectors r, .. r, of A, have the form of r1 ---Select--- r2 -Select--- ---Select--- and so forth. Then, the equation c,r, + c,r, + ... + cy = 0 implies which of the following equations? (Select all that apply.) U cze3n + C2e3n + Cze3n = 0 C1@2m + Cz€2m = 0 = 0 C1ein + Cz@2n + cze3n Cje im * Cze2m = 0 Cze 3n = 0 Cje11 = 0 You can conclude in turn that c, = c, Ck and so the row vectors are linearly independent.

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