Linear Midterm 2 TRUE/FALSE

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

F

Each vector (*x*, *y*, *z*) in R inverse.

T

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

T

For every vector (*x*1,*x*2,...,*xn*) in R*n*, the vector

(−1) · (*x*1, *x*2, . . . , *xn*) is an additive inverse.

T

A vector whose components are all positive is called

a “positive vector”

F

If *s* and *t* are scalars and **x** and **y** are vectors in R*n*,

then (*s* + *t*)(**x** + **y**) = *s***x** + *t***y**.

F

For every vector **x** in R*n*, the vector 0**x** is the zero

vector of R*n*.

T

T

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

F

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

T

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

F

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

F

The zero vector in a vector space *V* is unique.

T

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

F

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

*s* are scalars such that *r***v** = *s***v**, then *r* = *s*.

T

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

F

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

T

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

T

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

T

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

F

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

F

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.

F

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

F

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

F

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

T

If *m* < *n*, then R*m* is a subspace of R*n*

F

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

T

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* .

F

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* .

F

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

F

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

T

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

F

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* .

T

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*.

F

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

T

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

F

Every vector space *V* has a finite spanning set.

F

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* .

F

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

T

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

F

If *m* 

F

Every vector space *V* possesses a unique minimal spanning set.

F

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

T

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

F

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

T

If the Wronskian of a set of functions is nonzero at some point *x*0 in an interval *I*, then the set of func- tions is linearly independent.

T

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.

T

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

T

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.

F

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.

F

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

F

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

F

A vector space *V* can have many different bases.

T

dim\[*Pn*(R)\] = dim\[R*n*\]

F

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

F

Five vectors in *P*3(R) must be linearly dependent.

T

Two vectors in *P*3(R) must be linearly independent

F

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

F

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* .

F

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

T

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

F