document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1174473>Question 1174473</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the following subsets of the vector space R4?<BR>\n" );
document.write( "S1={(1,0,1,1),(1,−1,1,1),(0,1,0,0),(1,0,1,0)}<BR>\n" );
document.write( "S2={(1,0,1,2),(1,0,0,0),(0,0,2,−1)}<BR>\n" );
document.write( "Which of the following statements is true?\r<BR>\n" );
document.write( "\n" );
document.write( "Select one:<BR>\n" );
document.write( " S2 is linearly independent\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " S1 is linearly independent\r<BR>\n" );
document.write( "\n" );
document.write( " S1 spans R4\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " S2 spans R4<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850643 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850643\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's analyze each set to determine the correct answer.\r<BR>\n" );
document.write( "\n" );
document.write( "**S1 = {(1, 0, 1, 1), (1, -1, 1, 1), (0, 1, 0, 0), (1, 0, 1, 0)}**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Linear Independence:**<BR>\n" );
document.write( "    * To check for linear independence, we can form a matrix with these vectors as rows (or columns) and calculate its determinant. If the determinant is non-zero, the vectors are linearly independent.<BR>\n" );
document.write( "    * Alternatively, we can check the rank of the matrix. If the rank of the matrix is equal to the number of vectors, then the vectors are linearly independent.<BR>\n" );
document.write( "    * The code provided shows that S1 is linearly dependent.<BR>\n" );
document.write( "* **Spanning R4:**<BR>\n" );
document.write( "    * For a set of vectors to span R4, it must have at least 4 linearly independent vectors.<BR>\n" );
document.write( "    * Since S1 is linearly dependent, it cannot span R4.\r<BR>\n" );
document.write( "\n" );
document.write( "**S2 = {(1, 0, 1, 2), (1, 0, 0, 0), (0, 0, 2, -1)}**\r<BR>\n" );
document.write( "\n" );
document.write( "* **Linear Independence:**<BR>\n" );
document.write( "    * To check for linear independence, we can form a matrix with these vectors as rows (or columns) and check its rank.<BR>\n" );
document.write( "    * The code provided shows that S2 is linearly independent.<BR>\n" );
document.write( "* **Spanning R4:**<BR>\n" );
document.write( "    * For a set of vectors to span R4, it must have at least 4 vectors.<BR>\n" );
document.write( "    * Since S2 has only 3 vectors, it cannot span R4.\r<BR>\n" );
document.write( "\n" );
document.write( "**Conclusion**\r<BR>\n" );
document.write( "\n" );
document.write( "* S1 is linearly dependent and does not span R4.<BR>\n" );
document.write( "* S2 is linearly independent and does not span R4.\r<BR>\n" );
document.write( "\n" );
document.write( "Therefore, the correct statement is:\r<BR>\n" );
document.write( "\n" );
document.write( "* **S2 is linearly independent**<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );