Final answer:To determine if the columns of the matrix A span R2, we need to check if every vector in R2 can be expressed as a linear combination of the columns of A. We can find the rank of A by row reducing the augmented matrix [A|0]. If the rank is equal to 2, the columns span R2.Explanation:To determine if the columns of the matrix A span R2, we need to check if every vector in R2 can be expressed as a linear combination of the columns of A. In other words, we need to check if there exists scalars such that the equation Ax = b has a solution for every b in R2.In this case, A = [2 1; 0 1; -3 -1]. To check if the columns span R2, we can find the rank of A. If the rank of A is equal to 2 (the number of columns), then the columns span R2. If the rank is less than 2, then the columns do not span R2.To find the rank of A, we can row reduce the augmented matrix [A|0]. If the reduced row echelon form of [A|0] has a pivot in every row, then the rank of A is equal to 2. Otherwise, the rank is less than 2....

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