Concept explainers
Mark each statement True or False. Justify each answer. (If true, cite appropriate facts or theorems. If false, explain why or give a counterexample that shows why the statement is not true in every case.) In parts (a)-(f), v1, ..., vp are
- a. The set of all linear combinations of v1, ..., vp is a vector space.
- b. If {v1, ..., vp−1} spans V, then S spans V.
- c. If {v1, ..., vp−1} is linearly independent, then so is S.
- d. If S is linearly independent, then S is a basis for V.
- e. If Span S = V, then some subset of S is a basis for V.
- f. If dim V = p and Span, S = V, then S cannot be linearly dependent.
- g. A plane in ℝ3 is a two-dimensional subspace.
- h. The nonpivot columns of a matrix are always linearly dependent.
- i. Row operations on a matrix A can change the linear dependence relations among the rows of A.
- j. Row operations on a matrix can change the null space.
- k. The rank of a matrix equals the number of nonzero rows.
- l. If an m × n matrix A is row equivalent to an echelon matrix U and if U has k nonzero rows, then the dimension of the solution space of Ax = 0 is m − k.
- m. If B is obtained from a matrix A by several elementary row operations, then rank B = rank A.
- n. The nonzero rows of a matrix A form a basis for Row A.
- ○. If matrices A and B have the same reduced echelon form, then Row A = Row B.
- p. If H is a subspace of ℝ3, then there is a 3 × 3 matrix A such that H = Col A.
- q. If A is m × n and rank A = m, then the linear transformation x ↦ Ax is one-to-one.
- r. If A is m × n and the linear transformation x ↦ Ax is onto, then rank A = m.
- s. A change-of-coordinates matrix is always invertible.
- t. If B = {b1, ..., bn} and C = {c1, ..., cn} are bases for a vector space V, then the jth column of the change-of-coordinates matrix is the coordinate vector [cj]B.
a.
To find: Whether the statement “The set of all linear combinations of
Answer to Problem 1SE
The statement is true.
Explanation of Solution
Here, the given vectors are
The span {
Thus, the linear combinations of
Hence, the statement is true.
b.
To find: Whether the statement “If
Answer to Problem 1SE
The statement is true.
Explanation of Solution
The set S is
It is given that the set
That is, every element in the vector space
Here, the set
The smaller set
If the vector space V is spanned by
Hence, the statement is true.
c.
To find: Whether the statement “If
Answer to Problem 1SE
The statement is false.
Explanation of Solution
If
Therefore, it does not imply that S is linearly independent.
Hence, the statement is false.
d.
To find: Whether the statement “If S is linearly independent then S is a basis for V” is true or false.
Answer to Problem 1SE
The statement is false.
Explanation of Solution
A set of vectors
1. The set of vectors
2. The set
It can be seen that the second condition is not satisfied.
Hence, the statement is false.
e.
To find: Whether the statement “If
Answer to Problem 1SE
The statement is true.
Explanation of Solution
It is given that the vector space V is spanned by S, which is nonzero set.
Suppose the set S is linearly independent then, the set S form a basis for V.
Suppose the set S is linearly dependent then, some subset of S linearly independent and which spans V.
That is, some subset of S form a basis for V.
Hence, the statement is true.
f.
To find: Whether the statement “If
Answer to Problem 1SE
The statement is true.
Explanation of Solution
It is given that
Here,
Which implies the
Hence, the statement is false.
g.
To find: Whether the statement “A plane in
Answer to Problem 1SE
The statement is false.
Explanation of Solution
Every plane in
Sometimes plane in
Hence, the statement is false.
h.
To find: Whether the statement “The non-pivot columns of a matrix are always linearly dependent” is true or false.
Answer to Problem 1SE
The statement is false.
Explanation of Solution
Consider the matrix
Here non-pivot columns are linearly independent.
Hence, the statement is false.
i.
To find: Whether the statement “Row operations on a matrix A can change the linear dependence relations among the rows of A” is true or false.
Answer to Problem 1SE
The statement is true.
Explanation of Solution
Row operations on matrix
Hence, the statement is true.
j.
To find: Whether the statement “Row operations on a matrix can change the null space” is true or false.
Answer to Problem 1SE
The statement is false.
Explanation of Solution
Row operations do not change the solution set of the system
Therefore, row operations do not change the null space.
Hence, the statement is false.
k.
To find: Whether the statement “The rank of a matrix equals the number of nonzero rows” is true or false.
Answer to Problem 1SE
The statement is false.
Explanation of Solution
The rank of a matrix A is the dimension of the column space of A.
The dimension of column space of A is the number of pivot columns in A.
Therefore, the rank of matrix equals the number of pivot columns.
Consider the matrix
The above matrix has 2 rows but rank of the matrix is 1.
Hence, the statement is false.
l.
To find: Whether the statement “If an
Answer to Problem 1SE
The statement is false.
Explanation of Solution
If U has k nonzero rows then,
According to the Rank Theorem, the rank of an
Hence, the statement is false.
m.
To find: Whether the statement “If B is obtained from a matrix A by several elementary row operations, then
Answer to Problem 1SE
The statement is true.
Explanation of Solution
Elementary row operations does not change the number of pivot columns and hence, does not change the rank of a matrix.
Therefore, the rank of matrix B will be same as the rank of matrix A.
Hence, the statement is true.
n.
To find: Whether the statement “The nonzero rows of a matrix A form a basis for Row A” is true or false.
Answer to Problem 1SE
The statement is false.
Explanation of Solution
To form a basis for A, the rows have to span A and should also be linearly independent.
The nonzero rows of a matrix A span Row A but that does not guarantee that they are linearly independent.
Hence, the statement is false.
o.
To find: Whether the statement “If matrices A and B have the same reduced echelon form, then
Answer to Problem 1SE
The statement is true.
Explanation of Solution
The nonzero rows of the echelon form of a matrix, form a basis for the row space of that matrix.
If the echelon form for two matrices is same, then the basis for the row spaces is also same.
Since, row spaces are vector spaces and if two vector spaces have same basis, then the vector spaces are same.
Hence, the statement is true.
p.
To find: Whether the statement “If H is a subspace of
Answer to Problem 1SE
The statement is true.
Explanation of Solution
If H is a zero, 1, 2, or 3 dimensional subspace of
The basis of H will then be in the column space of A.
Therefore,
Hence, the statement is true.
q.
To find: Whether the statement “If A is
Answer to Problem 1SE
The statement is false.
Explanation of Solution
Here the matrix A is linear transformation from
The transformation
Here, rank of the matrix is A thus, by the rank nullity theorem null space of
Hence, the statement is false.
r.
To find: Whether the statement “If A is
Answer to Problem 1SE
The statement is true.
Explanation of Solution
Here the matrix A is linear transformation from
If the transformation is onto then,
The rank of a matrix A is the dimension of the column space of A.
Therefore, the rank of A is m.
Hence, the statement is true.
s.
To find: Whether the statement “A change-of-coordinate matrix is always invertible” is true or false.
Answer to Problem 1SE
The statement is true.
Explanation of Solution
The columns of
The matrix
Hence, the statement is true.
t.
To find: Whether the statement “If
Answer to Problem 1SE
The statement is false.
Explanation of Solution
The jth column of the change-of-coordinates matrix
Hence, the statement is false.
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