Math 308 Midterm T/F

False

False.

True.

True, by definition of equivalent matrices.

True. For example, 2 4 −(1/2)R1 + R2 => R2~ 1 1 2 4 0 −1 And 2 4 1 1 R1 ⇔ R2~ 1 1 2 4 −2R1 + R2 right double arrow implies R2~ 1 1 0 2

False, by Theorem 1.5.

False, It could be inconsistent, and therefore have no solutions, as with the system x1+x2+x3+x4 +x5+x6+x7=0 x1+x2+x3 =1 x4 +x5 =1 x6+ x7=1

True, a free variable can take any value, and so there are infinitely many solutions.

True. If it is consistent, there will be at least one free variable, and hence infinitely many solutions.

False. For example, the system x1 + x2 = 0 x1 + x2 = 1 has no solutions. And the system x1 + x2 = 1 2x1 + 2x2 = 2 has infinitely many solutions.

False. Scalars may be any real number, such as c = −1.

False. The sum c1 + u of a scalar and a vector is undefined.

False. They do not point in opposite directions, as there does not exist c < 0 such that [1, −4, 5] = c * [−2,8, 10]

False. For example, the length of 2*[1, 0] = [2, 0] is 2, but the length of (−6)*[1, 0] = [−6, 0] is 6.

False, the zero vector can be included with any set of vectors which already span Rn.

False, since every column of A may be a zero column.

False

True

True, the span of a set of vectors can only increase (with respect to set containment) when adding a vector to the set.

False. Consider u1 = (0, 0, 0), u2 = (1, 0, 0), u3 = (0, 1, 0), and u4 = (0, 0, 1).

True. The span of {u1, u2, u3} will be a subset of the span of {u1, u2, u3, u4}.

True. Since u4 is a linear combination of {u1, u2, u3}, any vector in span{u1, u2, u3, u4} can be written as a linear combination of {u1, u2, u3}.

True, since u4 is in span{u1, u2, u3, u4}, but u4 not in span{u1, u2, u3}.

If {u1, u2, , um} is a set of vectors in Rn and n < m, then the set is linearly dependent.

The vectors are not scalar multiples of each other.

Any collection of vectors containing the zero vector must be linearly dependent.

False. For example, u = (1, 0) and v = (2, 0) are linearly dependent but do not span R2.

If m > n, then a set of m vectors in Rn is linearly dependent.

False. For example, A = [1 2 3] [0 0 0] has more columns than rows, but the columns are linearly dependent.

False. For example, if A = [1, 1] and b = [1, 0], then Ax = b has no solution.

True. If {u1, u2, u3} is linearly dependent, then the equation x1u1 + x2u2 + x3u3 = 0 has a nontrivial solution, and therefore so does x1u1 + x2u2 + x3u3 + x4u4 = 0.

True. If {u1, u2, u3, u4} is linearly independent, then the equation x1u1 + x2u2 + x3u3 + x4u4 = 0 has only the trivial solution, and therefore so does x1u1 + x2u2 + x3u3 = 0.

False. Consider u1 = (1, 0, 0), u2 = (0, 1, 0), u3 = (0, 0, 1), u4 = (0, 0, 0).

False. If u4 = x1u1 + x2u2 + x3u3, then x1u1 + x2u2 + x3u3 − u4 = 0, and since the coefficient of u4 is −1, {u1, u2, u3, u4} is linearly dependent.

False. Consider u1 = (1, 0, 0), u2 = (1, 0, 0), u3 = (1, 0, 0), u4 = (0, 1, 0).

False. Consider u1 = (1, 0, 0, 0), u2 = (0, 1, 0, 0), u3 = (0, 0, 1, 0), u4 = (0, 0, 0, 1).