A linear system with 5 variables and 3 equations always has infinitely many

A linear system with 5 variables and 3 equations

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A linear system with 5 variables and 3 equations always hasinfinitely many solutions. TRUEFALSE If a linear system of the formAx=b, withAan (n, m)-matrixandba vector inRn, has a solution, then this solution canbe written as a vector inRm. Letuibe vectors inRn.Then the linear system with aug-mented matrix [u1u2. . .un|u2+u3] has [0,1,1, . . . ,0]tas asolution. A homogeneous system is always consistent. The trivial solution is always a solution to a linear system. TRUEFALSE IfAx=0has only one solution, this must be the trivialsolution. (b)Cross the right box for the statements about linear independence and span.{u1,u2,u3,u4} ⊆R3spansR3. {u1,u2}= span{u1+u2,u1-u2} Ifu1andu2are linearly dependent, then there exists a scalarcRsuch thatu1=cu2oru2=cu1. TRUEFALSE If{u1,u2,u3}spansR3, then so does{u1,u2,u3,u4}, for anyvectoru4R3. (c)Cross the right box for the statements about linear systems and linear independence.IfAis an (n, m)-matrix andba vector inRnand the columnsofAare linearly independent, then the linear systemAx=bcannot have free variables. A set of one vector is always linearly independent. Ifu1,u2, andu3are pairwise linearly independent, then {u1,u2,u3}is also linearly independent.A homogeneous system with 3 variables and 3 equations hasexactly one solution. TRUEFALSE (d)Cross the right box for the statements about linear homomorphismsIfAis an (n, m)-matrix, then the linear homomorphism thatmaps a vectorutoAuhasRmas domain andRnas codomain.

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