Math 308

A homogeneous linear system may have exactly one non-trivial solution.

F

A linear system with more variables than equations must have infinitely many solutions.

F

A set of two vectors is linearly dependent if and only if one is a scalar multiple of the other.

T

The following matrix is in RREF:1 0 3 0 | 40 1 0 0 | 30 0 0 1 | 6

T

If A is a matrix with more columns than rows, then the columns of A form a linearly independent set.

F

Every vector in span{u1, u2, u3} is in span{u1, u2}.

F

If v is a linear combination of u1 and u2 then v is also a linear combination of u1, u2, u3.

T

If {u1, u2} is linearly independent and {u1, u2, u3} is linearly dependent, then u3 is in span{u1, u2}.

F

If m > n any set of m vectors in R^n spans R^n.

F

Any linearly independent set of n vectors in R^n spans R^n.

T

A system of 2013 linear equations in 2014 variables can have a unique solution.

F

If u1 is not in the span of {u2, u3}, then {u1, u2, u3} is linearly independent.

F

A system of equations with more variables than equations cannot have a unique solution.

T

If m < n, a set of m vectors in R^n cannot span R^n.

T

If m < n, any set of m vectors in R^n is linearly independent.

F

If the equation Ax = 0 has a unique solution, then the columns of A are linearly independent.

T

If {u1, u2, u3} is linearly dependent, then {u1, u2, u3, u4} is linearly dependent (where all vectors have the same dimension)

T

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)

F

If A is a matrix in reduced echelon form, then every column of A must have a leading one.

F

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.

T

If span{u1, u2, u3, u4} = span{v1, v2, v3} then u1 must be a linear combination of {v1, v2, v3}.

T

A system of equations with more variables than equations always has infinitely many solutions.

F

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.

T

A system of equations with more equations than variables has no solutions.

F

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.

F

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.

T

The set of solutions to a linear system with n variables is a subspace of R^n.

F