1 please 1. Mark each statement True or False. Justify each answer. (Iftrue, cite appropriate facts or theorems. If false, explain whyor give a counterexample that shows why the statement is nottrue in every case.a. Every matrix is row equivalent to a unique matrix inechelon form.b. Any system of n linear equations in n variables has atmost n solutions.odc. If a system of linear equations has two different solu-tions, it must have infinitely many solutions.pituloeodd. If a system of linear equations has no free variables, thennoilozit has a unique solution.be. If an augmented matrix [Ab] is transformed into[C d] by elementary row operations, then the equa-tions Ax = b and Cx = d have exactly the same solu-tion sets.f. If a system Ax = b has more than one solution, then sodoes the system Ax = 0.neto (2)sulav simag. If A is an m ×n matrix and the equation Ax = b isconsistent for some b, then the columns of A span Rm.h. If an augmented matrix [ A b] can be transformed byelementary row operations into reduced echelon form,then the equation Ax = b is consistent.i. If matrices A and B are row equivalent, they have thesame reduced echelon form..8vilj. The equation Ax = 0 has the trivial solution if and onlyif there are no free variables.ApresU .01k.1.If A is an m x n matrix and the equation Ax = b is con-sistent for every b in R", then A has m pivot columns.If an m x n matrix A has a pivot position in every row,then the equation Ax = b has a unique solution for eachb in Rm.Lorm. If an nxn matrix A has n pivot positions, then thereduced echelon form of A is the n x n identity matrix.n. If 3 x 3 matrices A and B each have three pivot posi-tions, then A can be transformed into B by elementaryrow operations. ni vmonil1601If A is an m x n matrix, if the equation Ax = b has atunleast two different solutions, and if the equation Ax = cis consistent, then the equation Ax = c has many solu-bitions.0.p. If A and B are row equivalent m x n matrices and if thecolumns of A span R", then so do the columns of B.Holloscibiniq. If none of the vectors in the set S = {V1, V2, V3} in R³ isa multiple of one of the other vectors, then S is linearlysandi 10 independent.medoso ni por.2022mstava dimeS.t.TRYIf {u, v, w} is linearly independent, then u, v, and w arenot in R2.In some cases, it is possible for four vectors to spanR³.If u and v are in Rm, then -u is in Span{u, v).storu. If u, v, and w are nonzero vectors in R2, then w is a linearScombination of u and v.V.aniv. If w is a linear combination of u and v in R", then u is alinear combination of v and w.moldy.w. Suppose that V₁, V2, and v3 are in R5, v₂ is not a multipleof V₁, and v3 is not a linear combination of v₁ and v2.Then {V1, V2, V3} is linearly independent.ni moldong all states bancoter strongx. A linear transformation is a function.mIf A is a 6 x 5 matrix, the linear transformation x →→ Axcannot map R5 onto R6.ad gniertez. If A is an m x n matrix with m pivot columns, then thelinear transformation x Ax is a one-to-one mapping.gniwollol am) Torborw grinim 10 m maidong sdt robizno2. Let a and b represent real numbers. Describe the possiblesolution sets of the (linear) equation ax = b. [Hint: Thenumber of solutions depends upon a and b.]3. The solutions (x, y, z) of a single linear equationni max+by+ cz =14form a plane in R³ when a, b, and c are not all zero. Constructmal sets of three linear equations whose graphs (a) intersect ina single line, (b) intersect in a single point, and (c) have no

Answered: Image /qna-images/answer/2c2a4ffe-7cfb-4177-932a-2924ddd4369d.jpg

1. Mark each statement True or False. Justify each answer. (If true, cite appropriate facts or theorems. If false, explain why or give a counterexample that shows why the statement is not true in every case. a. Every matrix is row equivalent to a unique matrix in echelon form. b. Any system of n linear equations in n variables has at most n solutions. odc. If a system of linear equations has two different solu- tions, it must have infinitely many solutions. indoe ntions has no free variables then 61 dosto