Linear Algebra Section 3.4-4.3 True/False

A set containing a single vector is linearly independent

False

A vector space must contain at least two vectors.

False

Every linearly dependent set contains the zero vector

False

For all vectors u, v, and w in 3-space, the vectors (u X v) X w and u X (v X w) are the same.

False

If u, v, and w are vectors in R3, where u is nonzero and u X v = u X w, then v = w.

False

In the vector space F(-infty, infty) any function who graph passes through the origin is a zero vector.

False

The general solution of the non-homogeneous linear system Ax = b can be obtained by adding b to the general solution of the homogeneous linear system Ax = 0.

False

The scalar triple product of u, v, and w determines a vector whose length is equal to the volume of the parallelepiped determined by u, v, and w.

False

The set of positive real numbers is a vector space if vector addition and scalar multiplication are the usual operation of addition and multiplication are real numbers.

False

The three polynomials x+1, -3x+2, and x are linearly independent

False

The vector equation of a plane can be determined from any point lying on the plane and a nonzero vector parallel to the plane.

False

Two subsets of a vector space V that span the same subspace of V must be equal.

False

A normal vector can be obtained by taking the cross product of two nonzero and non-colinear vector lying in the plane.

True

A vector is any element of a vector space

True

All solution vectors of the linear system Ax = b are orthogonal to the row vectors of the matrix A if and only if b^-> = 0^->.

True

For the standard unit vectors i, j, k, the vectors (i x j) x k and i x (j x k) are the same

True

If the set of vectors v_1, ..., v_n is a linearly dependent set of vectors, then kv_1, ..., kv_n is a linearly dependent for every scalar k.

True

If u and v are vectors in 3-space, then ||v x u|| is equal to the area of the parallelogram determined by u and v.

True

If u is a vector and k is a scalar such that ku = 0 then either k = 0 or u = 0

True

If x1 and x2 are two solutions of the non homogeneous linear system Ax = b, then x1 - x2 is a solution of the corresponding homogeneous linear system.

True

In every vector space the vectors (-1)u and -u are the same

True

Some linearly dependent sets of vectors do not contain the zero vector

True

The cross product of two nonzero vectors u and v is a nonzero vector if and only if u and v are not parallel

True

The points lying on a line through the origin in R^2 and R^3 are all scalar multiples of any nonzero vector on the line.

True

The set of vectors (v,kv) is linearly dependent for every scalar k.

True

The set of vectors {v1, v2, v3} is linearly independent, then {kv1, kv2, kv3} is also a linearly independent for every nonzero scalar k.

True

The vector equation of a line can be determined from any point lying on the line and a nonzero vector parallel to the line

True