Chapter 4

The standard method for producing a spanning set for Nul A sometimes fails to produce a basis for Nul A.

The statement is false because the method always produces an independent set.

If B is any echelon form of​ A, then the pivot columns of B form a basis for the column space of A.

The statement is false. The columns of an echelon form B of A are often not in the column space of A.

The dimension of the null space of A is the number of columns of A that are not pivot columns.

The statement is true. The dimension of Nul A equals the number of free variables in the equation Ax = 0.

If B is the standard basis for set of real numbers R Superscript n, then the​ B-coordinate vector of an x in set of real numbers R Superscript n is x itself.

The statement is true. The standard basis consists of the columns of the n x n identity matrix. So [x]B = x1 e1 + ... + xn en = x.

For an m x n matrix A, rank (A) + dim Nul (A) =

n

The column space of an m×n matrix​ A, written as​ Col(A), is

the set of all linear combinations of the columns of A. If A = ​[a1, ... , an​], then the following is true. ​Col(A) = ​Span{a1​, . . .​ , an​}

If dim V = n, and if S spans V, then in order for S to be a basis for V...

... S must both span V and have n elements in order to be a basis of V.

If B is in echelon​ form, the nonzero rows of B form ...

... a basis for the row space of A as well as for that of B.

The dimension of a vector space is ...

... the number of vectors in a basis for that vector​ space

If two matrices A and B are row​ equivalent, then ...

... then their row spaces are the same. If B is in echelon​ form, the nonzero rows of B form a basis for the row space of A as well as for that of B.

For an m×n ​matrix, Col A is a subspace of ...

... ℝ^m

The correspondence [x]B -> x is called the coordinate mapping.

False. x -> [x]B is the definition of the coordinate mapping.

The dimension of Nul(A) is equal to the number of _____________ in the equation Ax = 0

Free variables

To test if w is in​ Col(A), reduce​ [A w​] to an echelon form.

If the system is​ consistent, then w is in​ Col(A). The reduced row echelon form of the augmented matrix​ [A w​] can be used to find a solution to the equation Ax=w.

Let W be the set of all vectors of the form shown on the​ right, where b and c are arbitrary. Find vectors u and v such that W = Span{u, v}. Why does this show that W is a subspace of set of ℝ^3​?

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

If there exists a set {v1, ... , vp } that spans V, then dim V <= p.

True. Apply the Spanning Set Theorem to the set {v1, ... , vP{ and produce a basis for V. This basis will not have more than P elements in it, so V <= p

Let A be an m×n matrix. Then the following statements are equivalent.

a. For each b in set of real numbers ℝ^m​, the equation Ax=b has a solution. b. Each b in set of real numbers ℝ^m is a linear combination of the columns of A. c. The columns of A span set of real numbers ℝ^m. d. The matrix A has a pivot position in every row.

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 the following conditions are met:

​(i) The set B is a linearly independent set. (ii) The subspace spanned by B coincides with H. In other​ words, H = Span { b1, ..., bp}