04 Vector Spaces

Question 1

Vector space

Answer 1

A vector space is a nonempty set of vectors on which are defined to operations, called addition and multiplication by scalars.

Question 2

Subspace

Answer 2

A subspace of a vector space V is a subset H of V that has three properties:

• The zero vector of V is in H.
• H is closed under vector addition.
• H is closed under multiplication by scalars.

Question 3

The vector space R^2 is_____ of R^3

Answer 3

The vector space R^2 is not a subspace of R^3.

Question 4

If v₁, …, v_p are in a vector space V, then Span{v₁, …, v_p} is ____ of V.

Answer 4

If v₁, …, v_p are in a vector space V, then Span{v₁, …, v_p​} is a subspace of V.

Question 5

Null space

Answer 5

The null space of an m x n matrix A, written as Nul A, is the set of all solutions of the homogeneous equation Ax = 0. In set notation,

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

Question 6

Column space

Answer 6

The column space of an m x n matrix A, written as Col A, is the set of all linear combinations of the columns of A. If A = [a₁ … a_n], then

Col A = Span{a₁, …, a_n}

Question 7

The column space of an m x n matrix A is all of R^n if and only if ____.

Answer 7

The column space of an m x n matrix A is all of R^n if and only if the equation Ax = b has a solution for each b in R^m.

Question 8

Linear transformation

Answer 8

A linear transformation T from a vector space V into a vector space W is a rule that assigns to each vector x in V a unique vector T(x) in W, such that

• T(u + v) = T(u) + T(v)
  for all u, v in V
• T(cu) = cT(u)
  for all u in V and all scalars c

Question 9

An indexed set {v₁ ,…, v_p} of two or more vector, with v₁ != 0, is linearly dependent if and only if _____.

Answer 9

An indexed set {v₁ ,…, v_p} of two or more vector, with v₁ != 0, is linearly dependent if and only if some v_j (with j > 1) is a linear combination of the preceding vectors, v₁, .., v_j-1​.

Question 10

Basis

Answer 10

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

• B is a linearly independent set
• The subspace spanned by B coincides with H; that is,
  H = Span{b₁, …, b_p}

Question 11

The pivot columns of a matrix A form ____ for Col A.

Answer 11

The pivot columns of a matrix A form a basis for Col A.

Question 12

Suppose B = {b₁, …, b_n} is a basis for V and x is in V. The coordinates of x relative to the basis B (or the B-coordinates of x) are the weights c₁, …, c_n such that x = ____.

Answer 12

Suppose B = {b₁, …, b_n} is a basis for V and x is in V. The coordinates of x relative to the basis B (or the B-coordinates of x) are the weights c₁, …, c_n such that x = c₁b₁ + … + c_nb_n​.

Question 13

Let B = {b₁, …, b_n} be a basis for a vector space V. Then the _____ x |-> [x]_B is a one-to-one linear transformation from V onto R^n.

Answer 13

Let B = {b₁, …, b_n} be a basis for a vector space V. Then the coordinate mapping x |-> [x]_B​ is a one-to-one linear transformation from V onto R^n.

Question 14

dimension of a vector space

Answer 14

If V is spanned by a finite set, then V is said to be finite-dimensional, and the cimension of V, written as dim V, is the number of vectors in a basis for V. The dimension of the zero vector space {0} is defined to be zero. If V is not spanned by a finite set, then V is said to be infinite-dimensional.

Question 15

The dimension of Nul A

Answer 15

The dimension of Nul A is the number of free variables in the equaion Ax = 0.

Question 16

The dimension of Col A

Answer 16

The dimension of Col A is the number of pivot columns in A.

Question 17

Rank

Answer 17

The rank of A is the dimension of the column space of A.

Question 18

rank _ + dim _ = n

Answer 18

rank A + dim Nul A = n

Question 19

The dimensions of the column space and ____ of an m x n matrix A are equal.

Answer 19

The dimensions of the column space and the row space of an m x n matrix A are equal.