Construct a 3x3 matrix, not in echelon form, whose columns span . Show that the constructed matrix has the desired property.
1. Matrix Construction:
• Provide the elements of the 3x3 matrix, ensuring that its columns collectively span the entire vector space .
2. Echelon Form Consideration:
• Confirm that the constructed matrix is not in echelon form. Explain the criteria or properties that distinguish it from echelon form.
3. Justification:
• Demonstrate through steps or transformations why the matrix’s columns span . Discuss the linear independence and coverage of the entire space.
Ensure clarity in your response, addressing both the construction of the matrix and the reasoning behind its spanning property.