Linear Midterm 2 TRUE/FALSE

The vector (x, y) in R2 is the same as the vector (x, y, 0) in R3.

Each vector (x, y, z) in R inverse.

The solution set to a linear system of 4 equations and 6 unknowns consists of a collection of vectors in R6

For every vector (x1,x2,...,xn) in Rn, the vector

(−1) · (x1, x2, . . . , xn) is an additive inverse.

A vector whose components are all positive is called

a “positive vector”

If s and t are scalars and x and y are vectors in Rn,

then (s + t)(x + y) = sx + ty.

For every vector x in Rn, the vector 0x is the zero

vector of Rn.

If x is a vector in the first quadrant of R2, then any scalar multiple kx of x is still a vector in the first quad- rant of R2.

The vector 5i−6j+√2k in R3 is the same as (5, −6, √2)

Three vectors x, y, and z in R3 always determine a 3-dimensional solid region in R3.

If x and y are vectors in R2 whose components are even integers and k is a scalar, then x + y and kx are also vectors in R2 whose components are even integers.

The zero vector in a vector space V is unique.

If v is a vector in a vector space V and r and s are scalars such that rv = sv, then r = s.

If v is a nonzero vector in a vector space V, and r and

s are scalars such that rv = sv, then r = s.

The set Z of integers, together with the usual oper- ations of addition and scalar multiplication, forms a vector space.

If x and y are vectors in a vector space V , then the additive inverse of x + y is (−x) + (−y)

The additive inverse of a vector v in a vector space V is unique.

The set {0}, with the usual operations of addition and scalar multiplication, forms a vector space

The set {0, 1}, with the usual operations of addition and scalar multiplication, forms a vector space.

The set of positive real numbers, with the usual op- erations of addition and scalar multiplication, forms a vector space.

The set of nonnegative real numbers (i.e., {x ∈ R : x ≥ 0}), with the usual operations of addition and scalar multiplication, forms a vector space.

The null space of an m × n matrix A with real elements is a subspace of Rm .

The solution set of any linear system of m equations in n variables forms a subspace of Cn

The points in R2 that lie on the line y=mx+b form a subspace of R2 if and only if b=0.

If m < n, then Rm is a subspace of Rn

A nonempty subset S of a vector space V that is closed under scalar multiplication contains the zero vector of V .

If V = R is a vector space under the usual operations of addition and scalar multiplication, then the subset R+ of positive real numbers, together with the oper- ations defined in Problem 20 of Section 4.2, forms a subspace of V .

If V = R3 and S consists of all points on the x y -plane, the xz-plane, and the yz-plane, then S is a subspace of V .

If V is a vector space, then two different subspaces of V can contain no common vectors other than 0.

The linear span of a set of vectors in a vector space V forms a subspace of V

If some vector v in a vector space V is a linear com- bination of vectors in a set S, then S spans V .

If S is a spanning set for a vector space V and W is a subspace of V , then S is a spanning set for W .

If S is a spanning set for a vector space V , then every vector v in V must be uniquely expressible as a linear combination of the vectors in S.

A set S of vectors in a vector space V spans V if and only if the linear span of S is V.

The linear span of two vectors in R3 must be a plane through the origin.

Every vector space V has a finite spanning set.

If S is a spanning set for a vector space V , then any proper subset S′ of S (i.e., S′ ̸= S) not a spanning set for V .

The vector space of 3 × 3 upper triangular matrices is spanned by the matrices Ei j where 1 ≤ i ≤ j ≤ 3.

A spanning set for the vector space P2(R) must contain a polynomial of each degree 0,1, and 2.

If m < n, then any spanning set for Rn must contain more vectors than any spanning set for Rm .

Every vector space V possesses a unique minimal spanning set.

The set of column vectors of a 5 × 7 matrix A must be linearly dependent.

The set of column vectors of a 7 × 5 matrix A must be linearly independent.

Any nonempty subset of a linearly independent set of vectors is linearly independent.

If the Wronskian of a set of functions is nonzero at some point x0 in an interval I, then the set of func- tions is linearly independent.

If it is possible to express one of the vectors in a set S as a linear combination of the others, then S is a linearly dependent set.

If a set of vectors S in a vector space V contains a linearly dependent subset, then S is itself a linearly dependent set.

A set of three vectors in a vector space V is linearly dependent if and only if all three vectors are proportional to one another.

If the Wronskian of a set of functions is identically zero at every point of an interval I, then the set of functions is linearly dependent.

A basis for a vector space V is a set S of vectors that spans V .

If V and W are vector spaces of dimensions n and m, respectively, and if n > m, then W is a subspace of V

A vector space V can have many different bases.

dim[Pn(R)] = dim[Rn]

If V is an n-dimensional vector space, then any set S of m vectors with m>n must span V

Five vectors in P3(R) must be linearly dependent.

Two vectors in P3(R) must be linearly independent

The set of all solutions to any nth order linear differ- ential equation forms an n-dimensional vector space.

If V is an n-dimensional vector space, then every set S with fewer than n vectors can be extended to a basis for V .

Every set of vectors that spans a finite-dimensional vector space V contains a subset which forms a basis for V

The set of all 3×3 upper triangular matrices forms a 3-dimensional subspace of M3(R)