Find Reduced Row Echelon Form [[s-2t],[3+3s],[3s+t],[2s]]

Problem

[[s−2*t],[3+3*s()],[3*s+t],[2*s()]]

Solution

1. Identify the nature of the expression. The given vector contains variables s() and t along with a constant term. To find a Reduced Row Echelon Form (RREF), we must represent the system as an augmented matrix where the columns correspond to the coefficients of s() t and the constant terms.

2. Rewrite the vector as a system of linear expressions in the form a*s+b*t=c

1*s−2*t=0

3*s+0*t=−3

3*s+1*t=0

2*s+0*t=0

3. Construct the augmented matrix [A|b] using the coefficients of s() and t as the columns and the constants as the right-hand side.

[[1,−2,0],[3,0,−3],[3,1,0],[2,0,0]]

4. Perform row operations to reach RREF. Start by multiplying the fourth row by 1/2 to get a leading 1

[[1,−2,0],[3,0,−3],[3,1,0],[1,0,0]]

5. Eliminate entries in the first column using the fourth row.

(R_1)←(R_1)−(R_4)

(R_2)←(R_2)−3*(R_4)

(R_3)←(R_3)−3*(R_4)

[[0,−2,0],[0,0,−3],[0,1,0],[1,0,0]]

6. Swap rows to organize the leading ones. Move (R_4) to the top and (R_3) to the second row.

[[1,0,0],[0,1,0],[0,−2,0],[0,0,−3]]

7. Eliminate the entry in the third row using the second row.

(R_3)←(R_3)+2*(R_2)

[[1,0,0],[0,1,0],[0,0,0],[0,0,−3]]

8. Normalize the fourth row by multiplying by −1/3

[[1,0,0],[0,1,0],[0,0,0],[0,0,1]]

9. Finalize the RREF by moving the zero row to the bottom.

[[1,0,0],[0,1,0],[0,0,1],[0,0,0]]

Final Answer

RREF*([s−2*t],[3+3*s()],[3*s+t],[2*s()])=[[1,0,0],[0,1,0],[0,0,1],[0,0,0]]

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