Question

Determine if the columns of the matrix form a linearly independent set. Justify your answer.

Answer 1

To determine if the columns of a matrix form a linearly independent set, we need to check if the only solution to the equation Ax = 0 is the trivial solution, where x is a vector of coefficients and 0 is a vector of zeros. If the only solution is the trivial solution, then the columns of the matrix are linearly independent. If there is a non-trivial solution, then the columns of the matrix are linearly dependent.

Without knowing the matrix in question, I cannot provide a specific answer to this question. However, the process for determining linear independence is as described above.

Answer 2

The columns of a matrix are linearly independent if there is only a trivial solution to the homogeneous system. A test of independence determines if two variables are related, using a contingency table and the chi-square statistic. The degrees of freedom are crucial for assessing statistical significance.

Step-by-step explanation:

The student's question pertains to determining whether the columns of a matrix form a linearly independent set. To justify whether they are linearly independent or not, one would generally perform operations such as calculating the determinant or using row reduction to get the matrix into reduced row-echelon form. If there exists a nontrivial solution to the homogeneous system (where all the solutions are not zeros), then the columns are linearly dependent. However, if only the trivial solution exists, the columns are linearly independent.

In statistics, a test of independence is used to decide if there is a relationship between two categorical variables. This often involves constructing a contingency table and calculating the chi-square statistic. The null hypothesis usually states that the variables are independent. If the chi-square calculated exceeds the critical value from the chi-square distribution table, the null hypothesis is rejected, suggesting that the variables are not independent.

Contingency Table and Test of Independence

A contingency table helps in the calculation of expected frequencies, and each cell must have an expected value of at least five to use this test. The number of degrees of freedom is calculated as (number of columns-1)(number of rows-1). This value is then used to assess the statistical significance of the observed chi-square statistic.