Suppose the reduced row echelon form of the matrix A is [1 0 0 0 0 1 0 0 0 0 1 0

Question

Suppose the reduced row echelon form of the matrix A is [1 0 0 0 0 1 0 0 0 0 1 0]. Which of the following must be true? (Circle the statements that must be true.) The columns of A span R^4 The transformation T(x) = Ax is one-to-one The transformation T(x) = Ax is onto. The equation Ax = 0 has no solution. The columns of A are linearly independent. The equation Ax = 0 has only the trivial solution. The equation Ax = b has a unique solution for each b in R^4 The equation Ax = b has at least one solution for each b in R^4

Explanation / Answer

In linear algebra, a matrix is in echelon form if it has the shape resulting from a Gaussian elimination. Row echelon form means that Gaussian elimination has operated on the rows and column echelon form means that Gaussian elimination has operated on the columns. In other words, a matrix is in column echelon form if its transpose is in row echelon form. The similar properties of column echelon form are easily deduced by transposing all the matrices.

All are true except d & h statement.

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