Question

Determine whether the set of vectors is a basis for ℛ3. Given the set of vectors , decide which of the following statements is true: A: Set is linearly independent and spans ℛ 3. Set is a basis for ℛ 3. B: Set is linearly independent but does not span ℛ 3. Set is not a basis for ℛ 3. C: Set spans ℛ 3 but is not linearly independent. Set is not a basis for ℛ 3. D: Set is not linearly independent and does not span ℛ 3. Set is not a basis for ℛ 3.

Answer 1

Answer:

(A) Set A is linearly independent and spans $R^3$. Set is a basis for $R^3$.

Step-by-Step Explanation

Definition (Linear Independence)

A set of vectors is said to be linearly independent if at least one of the vectors can be written as a linear combination of the others. The identity matrix is linearly independent.

Definition (Span of a Set of Vectors)

The Span of a set of vectors is the set of all linear combinations of the vectors.

Definition (A Basis of a Subspace).

A subset B of a vector space V is called a basis if: (1)B is linearly independent, and; (2) B is a spanning set of V.

Given the set of vectors  $A= \left(\begin{array}{[c][c][c][c]}1 & 0 & 0 & 0\\ 0 & 1 & 0 & 1\\ 0 & 0 & 1 & 1\end{array} \right)$ , we are to decide which of the given statements is true:

In Matrix $A= \left(\begin{array}{[c][c][c][c]}(1) & 0 & 0 & 0\\ 0 & (1) & 0 & 1\\ 0 & 0 & (1) & 1\end{array} \right)$ , the circled numbers are the pivots. There are 3 pivots in this case. By the theorem that The Row Rank=Column Rank of a Matrix, the column rank of A is 3. Thus there are 3 linearly independent columns of A and one linearly dependent column. $R^3$ has a dimension of 3, thus any 3 linearly independent vectors will span it. We conclude thus that the columns of A spans $R^3$.

Therefore Set A is linearly independent and spans $R^3$. Thus it is basis for $R^3$.