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I am reading Bruce Cooperstein's book: Advanced Linear Algebra ... ...
I am focused on Section 1.6 Bases and Finite-Dimensional Vector Spaces ...
I need help with the proof of Theorem 1.16 ...
Theorem 1.16 and its proof reads as follows:
View attachment 5134
Question 1
In the second paragraph of above proof we read the following:
"... ... Since $$y_1 \neq 0$$ it follows ... ... Can someone please explain exactly why $$y_1 \neq 0$$?Question 2
In the second paragraph of above proof we read the following:
"... ... By reordering the vectors of $$X$$, we can assume that $$x_n$$ is a linear combination of $$Z_1 = ( y_1, x_1, \ ... \ ... \ , x_{n-1}$$. ... ... "
Can someone explain (in some detail) exactly why $$x_n$$ is a linear combination of $$Z_1 = ( y_1, x_1, \ ... \ ... \ , x_{n-1}) $$?
Hope someone can help ... ...
Help will be much appreciated ... ...
Peter
NOTE: The text from Cooperstein displayed above mentions Theorem 1.14 ... so I am providing the statement of Theorem 1.14 as follows:View attachment 5135
I am focused on Section 1.6 Bases and Finite-Dimensional Vector Spaces ...
I need help with the proof of Theorem 1.16 ...
Theorem 1.16 and its proof reads as follows:
View attachment 5134
Question 1
In the second paragraph of above proof we read the following:
"... ... Since $$y_1 \neq 0$$ it follows ... ... Can someone please explain exactly why $$y_1 \neq 0$$?Question 2
In the second paragraph of above proof we read the following:
"... ... By reordering the vectors of $$X$$, we can assume that $$x_n$$ is a linear combination of $$Z_1 = ( y_1, x_1, \ ... \ ... \ , x_{n-1}$$. ... ... "
Can someone explain (in some detail) exactly why $$x_n$$ is a linear combination of $$Z_1 = ( y_1, x_1, \ ... \ ... \ , x_{n-1}) $$?
Hope someone can help ... ...
Help will be much appreciated ... ...
Peter
NOTE: The text from Cooperstein displayed above mentions Theorem 1.14 ... so I am providing the statement of Theorem 1.14 as follows:View attachment 5135