hkus10 Posted March 5, 2011 Share Posted March 5, 2011 (edited) 1.) The set W of all 2x3 matrices of the form a b c a 0 0 where c = a + b, is a subspace of M23 (Matrics 23). Show that every vector in W is a linear combination of W1 = 1 0 1 1 0 0 W2 = 0 1 1 0 0 0 Do I have to combine both W1 and W2 into one equation? Edited March 5, 2011 by hkus10 Link to comment Share on other sites More sharing options...
Cap'n Refsmmat Posted March 6, 2011 Share Posted March 6, 2011 Yes, that'd work. You can start with: [math]a \begin{pmatrix}1 & 0 & 1\\ 1 & 0 & 0\end{pmatrix} + b \begin{pmatrix}0 & 1 & 1\\ 0 & 0 & 0\end{pmatrix}[/math] and see where that leads you. Link to comment Share on other sites More sharing options...
hkus10 Posted March 6, 2011 Author Share Posted March 6, 2011 (edited) Yes, that'd work. You can start with: [math]a \begin{pmatrix}1 & 0 & 1\\ 1 & 0 & 0\end{pmatrix} + b \begin{pmatrix}0 & 1 & 1\\ 0 & 0 & 0\end{pmatrix}[/math] and see where that leads you. What I get is [math]\begin{pmatrix}a & b & 2c\\ a & 0 & 0\end{pmatrix}[/math] which is not [math]\begin{pmatrix}a & b & c\\ a & 0 & 0\end{pmatrix}[/math] Edited March 6, 2011 by hkus10 Link to comment Share on other sites More sharing options...
Cap'n Refsmmat Posted March 6, 2011 Share Posted March 6, 2011 You've made an error in that upper-right matrix element, then. Link to comment Share on other sites More sharing options...
hkus10 Posted March 6, 2011 Author Share Posted March 6, 2011 (edited) You've made an error in that upper-right matrix element, then. Is this the answer? aW_1 + bW_2 = [math] \begin{pmatrix}a & b & a+b\\ a & 0 & 0\end{pmatrix} [/math] where a, b can be any real number. Edited March 6, 2011 by hkus10 Link to comment Share on other sites More sharing options...
DrRocket Posted March 6, 2011 Share Posted March 6, 2011 What I get is [math]\begin{pmatrix}a & b & 2c\\ a & 0 & 0\end{pmatrix}[/math] How did you get an expression involving c starting from an expression involving only a and b ? You cannot do that without some other expression relating a and b to c. Link to comment Share on other sites More sharing options...
Cap'n Refsmmat Posted March 6, 2011 Share Posted March 6, 2011 The problem stipulates "where c = a + b". hkus10: Yes. And a + b = c, so the upper-right cell is equivalent to c. Not 2c. Link to comment Share on other sites More sharing options...
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