A homogeneous linear system may have exactly one non-trivial solution.
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A linear system with more variables than equations must have infinitely many solutions.
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A set of two vectors is linearly dependent if and only if one is a scalar multiple of the other.
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The following matrix is in RREF:1 0 3 0 | 40 1 0 0 | 30 0 0 1 | 6
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If A is a matrix with more columns than rows, then the columns of A form a linearly independent set.
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Every vector in span{u1, u2, u3} is in span{u1, u2}.
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If v is a linear combination of u1 and u2 then v is also a linear combination of u1, u2, u3.
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If {u1, u2} is linearly independent and {u1, u2, u3} is linearly dependent, then u3 is in span{u1, u2}.
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If m > n any set of m vectors in R^n spans R^n.
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Any linearly independent set of n vectors in R^n spans R^n.
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A system of 2013 linear equations in 2014 variables can have a unique solution.
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If u1 is not in the span of {u2, u3}, then {u1, u2, u3} is linearly independent.
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A system of equations with more variables than equations cannot have a unique solution.
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If m < n, a set of m vectors in R^n cannot span R^n.
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If m < n, any set of m vectors in R^n is linearly independent.
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If the equation Ax = 0 has a unique solution, then the columns of A are linearly independent.
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If {u1, u2, u3} is linearly dependent, then {u1, u2, u3, u4} is linearly dependent (where all vectors have the same dimension)
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If u4 is not a linear combination of {u1, u2, u3}, then {u1, u2, u3, u4} is linearly independent (where all vectors have the same dimension)
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If A is a matrix in reduced echelon form, then every column of A must have a leading one.
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If {u1, u2, u3} is a linearly independent set of vectors, and c1, c2, c3 are non-zero constants, then {c1u1, c2u2, c3u3} is also linearly independent.
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If span{u1, u2, u3, u4} = span{v1, v2, v3} then u1 must be a linear combination of {v1, v2, v3}.
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A system of equations with more variables than equations always has infinitely many solutions.
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If {v1, . . . , Vp} is a basis for a subset W of R^n, then any vector x in W can be written uniquely as a linear combintion of v1, ..., vp.
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A system of equations with more equations than variables has no solutions.
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If W is a subspace of R^n, and B = {v1, . . . , vn} is a basis for R^n, then some subset of the vectors in B form a basis for W.
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If S = {v1, v2, v3} is a linearly independent set of vectors in R^3, then any vector v in R^3 can be written as a linear combination of the vectors in S.
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The set of solutions to a linear system with n variables is a subspace of R^n.
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