Question

In Exercises 21 and 22, mark each statement True or False.\(H = Span\left\{ {{b_1},...,{b_p}} \right\}\)Justify

each answer.

21. a.A single vector by itself is linearly dependent.

b. If , then\(\left\{ {{b_1},...,{b_p}} \right\}\)is a basis for H.

c.The columns of an invertible\(n \times n\)matrix form a basis for\({\mathbb{R}^n}\).

d. A basis is a spanning set that is as large as possible.

e. In some cases, the linear dependence relations among the columns of a matrix can be affected by certain elementary row operations on the matrix.

Short answer

a. The statement is false.

b. The statement is false.

c. The statement is true.

d. The statement is false.

e. The statement is false.

Step-by-step solution

Mark the first statement true or false

The vectors are said to be linearly dependent if the equation \({c_1}{{\bf{v}}_1} + {c_2}{{\bf{v}}_2} + ... + {c_p}{{\bf{v}}_p} = 0\) has a non-trivial solution, where \({c_1},{c_2},...,{c_p}\) are scalars, and not all the weights are zero.

For a single vector, the equation can be written as \(c{\bf{v}} = 0\).

According to the given statement, the equation should be linearly dependent, then the equation \(c{\bf{v}} = 0\) should have a non-trivial solution. It means weight \(c\) is non-zero and vector v is also non-zero.

As \(c\) and v are non-zero, so the equation \(c{\bf{v}} = 0\) cannot be satisfied. It is true only for zero vector.

So, the statement “A single vector by itself is linearly dependent.” is not always true.

Thus, statement (a) is false.

(b) Step 2: Mark the second statement true or false

The given statement “If\(H = {\rm{Span}}\left\{ {{{\bf{b}}_1},...,{{\bf{b}}_p}} \right\}\), then\(\left\{ {{{\bf{b}}_1},...,{{\bf{b}}_p}} \right\}\)is a basis for H.” can’t be true the reason is as follows:

If\(H = {\rm{Span}}\left\{ {{{\bf{b}}_1},...,{{\bf{b}}_p}} \right\}\), then the set of vectors\(\left\{ {{{\bf{b}}_1},...,{{\bf{b}}_p}} \right\}\)can be linearly dependent or independent.But\(\left\{ {{{\bf{b}}_1},...,{{\bf{b}}_p}} \right\}\) is the basis for H then the vectors are linearly independent.

Thus, statement (b) is false.

(c) Step 3: Mark the third statement true or false

Recall the invertible matrix theorem, which states that if the inverse of the matrix exists or the matrix is invertible, then the columns are linearly independent, and they form a basis for\({\mathbb{R}^n}\).

Thus, statement (c) is true.

Mark the fourth statement true or false

The given statement “A basis is a spanning set that is as large as possible.” cannot always be true.

The number of spanning sets can be more than the number of basis sets in\({\mathbb{R}^n}\). Not every spanning set can be used as a basis.

Thus, statement (d) is false.

(e) Step 5: Mark the fifth statement true or false

Consider matrices A and B, where matrix B is the row-reduced form of matrix A.

Matrix A can be different from matrix B, but the solution set of both the equations\(A{\bf{x}} = 0\)and\(B{\bf{x}} = 0\)will be the same.

So,the linear dependence relations among the columns of a matrix cannot be affected by certain elementary row operations on the matrix.

Thus, statement (e) is false.