Math 54 Midterm 2

Front 1

vector space

Back 1

a nonempty set V of objects, called vectors, on which are defined two operations, called addition and multiplication by scalars (real numbers), subject to the ten axioms

Front 2

ten axioms of a vector space (must hold for all vectors u, v, and w in V for all scalars c and d)

Back 2

(1) the sum of u and v, denoted by u+v, is in V

(2) u+v=v+u

(3) (u+v)+w=u+(v+w)

(4) There is a zero vector 0 in V such that u+0=u

(5) For each u in V, there is a vector -u in V such that u+(-u)=0

(6) The scalar multiple of u by c, denoted by cu, is in V

(7) c(u+v)=cu+cv

(8) (c+d)u=cu+du

(9) c(du)=(cd)u

(10) 1u=u

Front 3

What is the vector -u?

Back 3

the negative of the vector u that is unique for each u in V

Front 4

For each vector u in V and scalar c...

Back 4

(1) 0u=0

(2) c0=0

(3) -u=(-1)u

Front 5

subspace of a vector space V (and its three properties)

Back 5

a subset H of V that has three properties:

(1) the zero vector of V is in H

(2) H is closed under vector addition (for each u and v in H, the sum u+v is in H)

(3) H is closed under multiplication by scalars (for each u in H and each scalar c, the vector cu is in H)

Front 6

True or False: Every subspace is a vector space?

Back 6

True. Every subspace is a vector space

Front 7

zero subspace

Back 7

set consisting of only the zero vector in a vector space V; is a subspace; written as {0}

Front 8

True of False: A plane in R3 not through the origin is not subspace of R3

Back 8

True, because it does not contain the zero vector of R3

Front 9

linear combination

Back 9

refers to any sum of scalar multiples of vectors, and Span{v1,..,vp} denotes the set of all vectors that can be written as linear combinations of v1,..,vp

Front 10

True or False: If v1,...,vp are in a vector space V, then Span{v1,...vp} is not a subspace of V

Back 10

False. It is a subspace.

Front 11

What is Span{v1,...vp}?

Back 11

the subspace spanned (or generated) by {v1,...,vp}

Front 12

spanning (or generating) set for H

Back 12

a set {v1,...,vp} in H such that H=Span{v1,...,vp}

Front 13

solution set

Back 13

the set of all x that satisfy the system of equations

Front 14

null space (of an mxn matrix A)

Back 14

written as Nul A, it is the set of all solutions of the homogeneous equations Ax=0

Nul A={x: x is in n and Ax=0}

Front 15

Is the null space of a matrix a vector space?

Back 15

Yes. The null space of an mxn matrix A is a subspace of n. Equivalently, the set of all solutions to a system Ax=0 of m homogeneous linear equations in n unknowns is a subspace of n

Front 16

What is the first step of finding a spanning set for the null space of matrix?

Back 16

The first step is to find the general solution of Ax=0 in terms of free variables

Front 17

What are two things that apply to all problems where Nul A contains nonzero vectors?

Back 17

(1) the spanning set produced is automatically linearly independent (because the free variables are the weights on the spanning vectors)

(2) when Nul A contains nonzero vectors, the number of vectors in the spanning set for Nul A equals the number of free variables in the equations Ax=0

Front 18

column space (of a mxn matrix A)

Back 18

written as Col A, it is the set of all linear combinations of the columns of A

If A=[a1 ... an], then Col A=Span{a1,...,an}

and

Col A={b: b=Ax for some x in n}

Front 19

True or False: column space of an mxn matrix A is a subspace of Rm

Back 19

True. The column space is a subspace

Front 20

True or False: Col A is the co-domain of the linear transformation x-->Ax

Back 20

False. Col A is the range of the linear transformation

The column space of mxn matric A is all of Rm if and only if the equation Ax=b has a solution for each b in Rm

Front 21

True or False: When a matrix is not square, the vectors in Nul A and Col A live in the same "universe"

Back 21

False. When a matrix is not square, the vectors in Nul A and Col A live in entirely different "universes" (for example, in a mxn matrix, then Col A is a subspace of Rm and Nul A is a subspace of Rn)

Front 22

General Information about Nul A

Back 22

(1) Nul A is a subspace of Rn

(2) Nul A is implicitly defined (you are given only the condition Ax=0 that vectors in Nul A must satisfy

(3) It takes time to find vectors in Nul A. Row operations on [A 0] are required.

(4) There is no obvious relation between Nul A and the entries of A

(5) A typical vector v in Nul A has the property that Av=0

(6) Given a specific vector v, it is easy to tell if v is in Nul A. Just compute Av

(7) Nul A={0} if and only if the equation Ax=0 has only the trivial solution

(8) Nul A={0} if and only if the linear transformation x-->Ax is one-to-one

Front 23

General Information about Col A

Back 23

(1) Col A is a subspace of Rm

(2) Col A is explicitly defined (you are told how to build vectors in Col A)

(3) It is easy to find vectors in Col A; the columns of A are displayed, others are formed from them

(4) There is an obvious relation between Col A and the entries in A, since each column of A is in Col A

(5) A typical vector v in Col A has the property that the equation Ax=v is consistent

(6) Given a specific vector v, it may take time to tell if v is in Col A; row operations on [A v] are required

(7) Col A=Rm if and only if the equation Ax=b has a solution for every b in Rm

(8) Col A=Rm if and only if the linear transformation x-->Ax maps Rn onto Rm

Front 24

What is a linear transformation T (from a vector space V into a vector space W)?

Back 24

it is a rule that assigns to each vector x in V a unique vector T(x) in W, such that:

(1) T(u+v)=T(u)+T(v) for all u,v in V

and

(2) T(cu)=cT(u) for all u in V and all scalars c

Front 25

kernel of linear transformation T

Back 25

the null space of T; the set of all u in V such that T(u)=0 (the zero vector in W); it is a subspace of V

Front 26

range of linear transformation T

Back 26

the set of all vectors in W of the form T(x) for some x in V; it is a subspace of W

if T happens to arise as a matrix transformation--say, T(x)=Ax for some matrix A--then the range of T is the just the column space of A

Front 27

linear independence

Back 27

an indexed set of vectors {v1,..,vp} in V that only has a trivial solution for the vector equation:

c1v1+c2v2+...+cpvp=0 where c1=0,...,cp=0

Front 28

linear dependence

Back 28

an indexed set of vectors {v1,..,vp} in V that has a nontrivial solution for the vector equation:

c1v1+c2v2+...+cpvp=0 where c1,...,cp aren't all 0

Front 29

True or False: a set containing a single vector v is linearly dependent if and only if v does not equal zero

Back 29

False. A set containing a single vector v is linearly independent if and only if v does not equal zero

Front 30

True or False: any set containing the zero vector is linearly dependent

Back 30

True (the zero vector is a multiple of any vector with scalar zero, and if the vectors are multiples of each other then it is linearly dependent).

Front 31

When is an indexed set {v1,..,vp} of two or more vectors, with v1 not equaling zero, linearly dependent?

Back 31

it's linearly dependent if and only if some vj (with j>1) is a linear combination of the preceding vectors, v1,..,vj-1

Front 32

basis (for subspace H of vector space V)

Back 32

Let H be a subspace of a vector space V. An indexed set of vectors B={b1,..,bp} in V is a basis for H if:

(1) B is a linearly independent set, an

(2) the subspace spanned by B coincides with H, so that H=Span{b1,..,bp}

Front 33

Does our definition of a basis apply to the case H=V?

Back 33

Yes, because any vector space is a subspace of itself.

Front 34

Suppose A is an invertible nxn matrix. Do the columns of A form a basis for Rn?

Back 34

Yes, because they are linearly independent and they span Rn, by the Invertible Matrix Theorem

Front 35

standard basis for Rn

Back 35

the set {e1,..,en} where e1,..,en are the columns of the nxn identity matrix, In

Front 36

What is the Spanning Set Theorem?

Back 36

Let S={v1,..,vp} be a set in V, and let H=Span{v1,..,vp}. Then:

(1) If one of the vectors in S (say, vk) is a linear combination of the remaining vectors in S, then the set formed from S by removing vk still spans H.

(2) If H doesn't equal {0}, some subset of S is a basis of H

Front 37

When matrix A is row reduced to matrix B, the columns of B are often totally different from the columns of A. Do they have the same linear dependence relationships?

Back 37

Yes. Despite having different columns, matrix A and matrix B have the same solution set, and both of their columns have the exact same linear dependence relationships.

Front 38

True or False: The pivot columns of matrix A form a basis for Col A.

Back 38

True. Every nonpivot column of A is a linear combination of the pivot columns of A. Thus the nonpivot columns of A may be discarded from the spanning set for Col A (by the Spanning Set Theorem). This leaves the pivot columns of A as a basis for Col A.

WARNING: The pivot columns of a matrix A are evident when A has been reduced only to echelon form. Be careful to use the pivot columns of A itself for the basis of Col A.

Front 39

True or False: A basis is a linearly independent set that is as large as possible.

Back 39

True. The deletion of vectors from a spanning set must stop when the set becomes linearly independent. If an additional vector is deleted, it will not be a linear combination of the remaining vectors, and the smaller set will no longer span V