(a) The set of all m × m symmetric matrices in the set of all m × m matrices.
(b) The set of all solutions of Ax = 0 in Rn
(c) The set of all linear combinations of vectors in {(1, 1, 0), (1, 0, 0)} ⊂ R3
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Determine if the following subsets are subspaces of the vector spaces. If they are, prove it. If not, justify your answer.
10. Det ermine whether the following subsets W are subspaces of the given vect or spaces: (a) The set of 2 2 matrices given by W. A є M2.2 : A- as a subset of V M2,2 (b) The set of all 3 x 3 upper triangular matrices as a subset of V-M33- (c) The subset of vect ors in R3 of the for (2+x3, r2, r3). (d) The subset of vect ors in R2 of the form (ri,0) (e)...
I need help with these linear algebra problems. 1. Consider the following subsets of R3. Explain why each is is not a subspace. (a) The points in the xy-plane in the first quadrant. (b) All integer solutions to the equation x2 + y2 = z2 . (c) All points on the line x + z = 5. (d) All vectors where the three coordinates are the same in absolute value. 2. In each of the following, state whether it is...
Let A be an m × n matrix, let x Rn and let 0 be the zero vector in Rm. (a) Let u, v є Rn be any two solutions of Ax 0, and let c E R. Use the properties of matrix-vector multiplication to show that u+v and cu are also solutions of Ax O. (b) Extend the result of (a) to show that the linear combination cu + dv is a solution of Ax 0 for any c,d...
(1 point) Determine whether the given set S is a subspace of the vector space V. A. V = R", and S is the set of solutions to the homogeneous linear system Ax = 0 where A is a fixed mxn matrix. B. V is the vector space of all real-valued functions defined on the interval (-oo, oo), and S is the subset of V consisting of those functions satisfying f(0) 0 C. V Mn (R), and S is the...
L. Answer True or False. Justify your answer (a) Every linear system consisting of 2 equations in 3 unknowns has infinitely many solutions (b) If A. B are n × n nonsingular matrices and AB BA, then (e) If A is an n x n matrix, with ( +A) I-A, then A O (d) If A, B two 2 x 2 symmetric matrices, then AB is also symmetric. (e) If A. B are any square matrices, then (A+ B)(A-B)-A2-B2 2....
In Exercises 3-4, use the Subspace Test to determine which of the sets are subspaces of Mnn. 3. a. The set of all diagonal n x n matrices. b. The set of all n × n matrices A such that det(A) = 0. c. The set of all n × n matrices A such that tr(A) = 0. d. The set of all symmetric n × n matrices.4. a. The set of all n × n matrices A such that AT = -A. b. The set...
linear algebra 2. Which of the following subsets of Rare actually subspaces? Justify your answer in terms of the definition and properties of subspaces. (a) The vectors [x y z]" with x + 2y -z = 0. (b) The vectors [a b c]" with a + b + c = 3. (c) The vectors [a+2bb-3b]' where a, b are any real numbers, (d) The vectors [pr] where q.r are any real numbers and p20.
Are the following subsets subspaces of the given vector space?Justify your answers using words and proper mathematical notation. If the set is not a subspace of the given vector space, give a counterex- ample (an example that demonstrates that one of the axioms fails) and explain why this shows the subset is not a subspace. If the set is a subspace, then prove it by showing that the conditions for a subset to be a subspace are met (a) S...
Problem 6-20 points. This question is about vector spaces and subspaces. (a) Define the terms "vector space" and "subspace" as precisely as you can. (b) Consider a line through the origin in R2, for example, the r-axis. Explain why this line is, or is not, a subspace of R2 in terms of your definitions in (a). (c) Consider the union of two lines through the origin in R2, for example, the z- and y-axes. Explain why this union of lines...
Mark each statement as True or False and justify your answer. a) The columns of a matrix A are linearly independent, if the equation Ax = 0 has the trivial solution. b) If vi, i = 1, ...,5, are in RS and V3 = 0, then {V1, V2, V3, V4, Vs} is linearly dependent. c) If vi, i = 1, 2, 3, are in R3, and if v3 is not a linear combination of vi and v2, then {V1, V2,...