If {u1, u2, u3, u4} does not span R3, then neither does {u1, u2, u3}.
answer
True. The span of {u1, u2, u3} will be a subset of the span of {u1, u2, u3, u4}.
question
If u4 is a linear combination of {u1, u2, u3}, then
span{u1, u2, u3, u4} = span{u1, u2, u3}.
answer
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}.
question
If u4 is not a linear combination of {u1, u2, u3}, then
span{u1, u2, u3, u4} β span{u1, u2, u3}.
answer
True, since u4 is in span{u1, u2, u3, u4}, but u4 not in span{u1, u2, u3}.
question
Linearly dependent.
answer
If {u1, u2, , um} is a set of vectors in Rn and n < m, then the set is linearly dependent.
question
Linearly independent.
answer
The vectors are not scalar multiples of each other.
question
Linearly dependent.
answer
Any collection of vectors containing the zero vector must be linearly dependent.
question
If a set of vectors in Rn is linearly dependent, then the set must span Rn.
answer
False. For example, u = (1, 0) and v = (2, 0) are linearly dependent but do not span R2.
question
Linearly dependent.
answer
If m > n, then a set of m vectors in Rn is linearly dependent.
question
If A is a matrix with more columns than rows, then the columns of A are linearly independent.
answer
False. For example, A = [1 2 3]
[0 0 0]
has more columns than rows, but the columns are linearly dependent.
question
If A is a matrix with linearly independent columns, then
Ax = b has a solution for all b.
answer
False. For example, if A = [1, 1] and b = [1, 0], then Ax = b has no solution.
question
If {u1, u2, u3} is linearly dependent, then so is {u1, u2, u3, u4}.
answer
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.
question
If {u1, u2, u3, u4} is linearly independent, then so is
{u1, u2, u3}.
answer
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.
question
If {u1, u2, u3, u4} is linearly dependent, then so is
{u1, u2, u3}.
If u4 is a linear combination of {u1, u2, u3}, then {u1, u2, u3, u4} is linearly independent.
answer
False. If u4 = x1u1 + x2u2 + x3u3, then x1u1 + x2u2 + x3u3 β u4 = 0, and since the coefficient of u4 is β1, {u1, u2, u3, u4} is linearly dependent.
question
If u4 is not a linear combination of {u1, u2, u3}, then {u1, u2, u3, u4} is linearly independent.