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I am reading C. G. Gibson's book: Elementary Geometry of Algebraic Curves.
I need some help with aspects of Example 1.4
The relevant text from Gibson's book is as follows:
View attachment 4562
Question 1In the above text, Gibson writes the following:
" ... ... Then a brief calculation verifies that any point \(\displaystyle p + t(q - p) = (1- t)p + tq\) also lies on the line ... ... "
I am unable to perform the brief calculation that Gibson refers to ... can someone please help me by showing the calculation and how it works ... ...
Question 2" ... Since at least one of a,b is non-zero, the system has a non-trivial solution, By linear algebra the \(\displaystyle 3 \times 3\) matrix of coefficients is singular, so the rows \(\displaystyle (p_1, p_2 , 1) , ( q_1, q_2, 1) , ( r_1, r_2, 1)\) are linearly independent. ... ... "
Can someone lease explain how we know that the \(\displaystyle 3 \times 3\) matrix of coefficients is singular?
Hope someone can help with the above two questions ...
Peter
I need some help with aspects of Example 1.4
The relevant text from Gibson's book is as follows:
View attachment 4562
Question 1In the above text, Gibson writes the following:
" ... ... Then a brief calculation verifies that any point \(\displaystyle p + t(q - p) = (1- t)p + tq\) also lies on the line ... ... "
I am unable to perform the brief calculation that Gibson refers to ... can someone please help me by showing the calculation and how it works ... ...
Question 2" ... Since at least one of a,b is non-zero, the system has a non-trivial solution, By linear algebra the \(\displaystyle 3 \times 3\) matrix of coefficients is singular, so the rows \(\displaystyle (p_1, p_2 , 1) , ( q_1, q_2, 1) , ( r_1, r_2, 1)\) are linearly independent. ... ... "
Can someone lease explain how we know that the \(\displaystyle 3 \times 3\) matrix of coefficients is singular?
Hope someone can help with the above two questions ...
Peter