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}
