MATH 2200 Exam 3 True or False

True

A normal vector to a plane can be obtained by taking the cross product of two nonzero and noncollinear vectors lying in the plane.

True

A vector is any element of a vector space.

False

A vector space must contain at least two vectors.

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.

False

Collinear vectors with the same length are equal.

False

Every subset of a vector space V that contains the zero vector in V is a subspace of V.

True

Every subspace of a vector space is itself a vector space.

False

Every vector in R^n has a positive norm.

True

Every vector space is a subspace of itself.

False

For all vectors u and v, it is true that || u + v || = || u || + || v ||

False

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

True

For all vectors u, v, and w in R^n, we have || u + v + w || ≤ || u || + || v || + || w ||

True

If (a, b, c) + (x, y, z) = (x, y, z), then (a, b, c) must be the zero vector.

True

If a and b are orthogonal vectors, then for every nonzero vector u, we have proba(projb(u)) = 0.

False

If a and b are scalars such that au + bv = 0, then u and v are parallel vectors.

True

If a and u are nonzero vectors, then PROJa(PROJa(u)) = PROJa(u).

True

If each component of a vector in R^3 is doubled, the norm of that vector is doubled.

False

If k and m are scalars and u and v are vectors, then (k + m)(u + v) = ku + mv

False

If k is a scalar and v is a vector, then v and kv are parallel if and only if k ≥ 0.

False

If the relationship PROJa(u) = PROJa(v) holds for some nonzero vector a, then u = v.

True

If the vectors v and w are given, then the vector equation 3(2v − x) = 5x − 4w + v can be solved for x

True

If u + v = u + w, then v = w.

True

If u and v are orthogonal vectors, then for all nonzero scalars k and m, ku and mv are orthogonal vectors.

False

If u is a vector and k is a scalar such that ku = 0, then it must be true that k = 0.

False

If u · v = 0, then either u = 0 or v = 0.

False

If u · v = u · w, then v = w.

False

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

True

If v is a nonzero vector in R^n, there are exactly two unit vectors that are parallel to v.

True

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

True

If || u ||= 2, || v || = 1, and u · v = 1, then the angle between u and v is π/3 radians.

True

In R^2, if u lies in the first quadrant and v lies in the third quadrant, then u · v cannot be positive.

True

In R^2, the vectors of norm 5 whose initial points are at the origin have terminal points lying on a circle of radius 5 centered at the origin.

True

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

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.

False

The expressions (u · v) + w and u · (v + w) are both meaningful and equal to each other.

False

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

True

The intersection of any two subspaces of a vector space V is a subspace of V.

False

The linear combinations a1v1 + a2v2 and b1v1 + b2v2 can only be equal if a1 = b1 and a2 = b2.

True

The orthogonal projection of u on a is perpendicular to the vector component of u orthogonal to a.

True

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

False

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

False

The solution set of a consistent linear system Ax = b of m equations in n unknowns is a subspace of R^n.

True

The span of any finite set of vectors in a vector space is closed under addition and scalar multiplication.

False

The union of any two subspaces of a vector space V is a subspace of V.

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.

False

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

True

The vectors (3, −1, 2) and (0, 0, 0) are orthogonal.

False

The vectors (a, b) and (a, b, 0) are equivalent.

True

The vectors v + (u + w) and (w + v) + u are the same.

False

Two equivalent vectors must have the same initial point.

False

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