The Aftermath
Linear Algebra of Maths
Consider the following linear equations:
x1 +3x2+9x3 +2x4 = f
x1+3x3−4x4=g
x2 +2x3 +3x4 =h
−2x1+3x2+5x4=k.
a . Find the coefficient matrix and the augmented matrix. Use the Gauss algorithm to find a row equivalent augmented matrix in echelon form; write at least three intermediate results (matrices) and annotate on the equivalent symbol ∼ what row elementary operations are used to get each intermediate matrix. Find the pivot columns and pivot positions in the original augmented matrix.
b . Continue the Gauss-Jordan algorithm but beginning with the rightmost pivot in the coefficient matrix part, i.e., skipping any pivot in the augment column, to find a row equivalent augmented matrix with the coefficient part in reduced echelon form.
c . Rewrite the row equivalent augmented matrix as the equivalent system of linear equations. When will the system be consistent and inconsistent? Find the solution set when the system is consistent. Write it in the parametric vector form.
d . Rewrite the original linear equations in vector form and in matrix form. For what value of k the vector [1]
[0]
[1]
[k]
is in the set spanned by the columns of the coefficient matrix?
e . Write the non-pivot column of the original coefficient matrix as a linear combination of the pivot columns of the original coefficient matrix.
