question

A linear system with three equations and two variables must be inconsistent.

answer

False

question

A linear system with three equations and five variables must be consistent.

answer

False.

question

A system in echelon form can have more variables than equations.

answer

True.

question

If two matrices are equivalent, then one can be transformed into the other with a sequence of elementary row operations.

answer

True, by definition of equivalent matrices.

question

Different sequences of row operations can lead to different echelon forms for the same matrix.

answer

True. For example,
2 4 ā(1/2)R1 + R2 => R2~
1 1
2 4
0 ā1
And
2 4
1 1 R1 ā R2~
1 1
2 4 ā2R1 + R2 right double arrow implies R2~
1 1
0 2

question

Different sequences of row operations can lead to different reduced echelon forms for the same matrix.

answer

False, by Theorem 1.5.

question

If a linear system has four equations and seven variables, then it must have infinitely many solutions.

answer

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

question

Every linear system with free variables has infinitely many solutions.

answer

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

question

Any linear system with more variables than equations cannot have a unique solution.

answer

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

question

If a linear system has the same number of equations and variables, then it must have a unique solution.

answer

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.

question

A vector can have positive or negative components, but a scalar must be positive.

answer

False. Scalars may be any real number, such as
c = ā1.

question

If c1 and c2 are scalars and u is a vector, then
(c1 + u)c2 = c1c2 + c2u.

answer

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

question

The vectors [1,ā4, 5] and [ā2, 8,10] point in opposite directions.

answer

False. They do not point in opposite directions, as there does not exist c < 0 such that [1, ā4, 5] = c * [ā2,8, 10]

question

The vector 2u is longer than the vector ā6u.

answer

False. For example, the length of 2*[1, 0] = [2, 0] is 2, but the length of (ā6)*[1, 0] = [ā6, 0] is 6.

question

If a set of vectors includes 0, then it cannot span
Rn.

answer

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

question

Suppose A is a matrix with n rows and m columns. If
n < m, then the columns of A span Rn.

answer

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

question

Suppose A is a matrix with n rows and m columns. If
m < n, then the columns of A span Rn.

answer

False

question

If A is a matrix with columns that span Rn, then
Ax = b has a solution for all b in Rn.

answer

True

question

If {u1, u2, u3} spans R3, then so does {u1, u2, u3, u4}.

answer

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

question

If {u1, u2, u3} does not span R3, then neither does {u1, u2, u3, u4}.

answer

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

question

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} ā 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}.

answer

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

question

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 ā u4 = 0, and since the coefficient of u4 is ā1, {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.

answer

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

question

If u4 is not a linear combination of {u1, u2, u3}, then {u1, u2, u3, u4} is linearly dependent.

answer

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