Answer with Step-by-step explanation:
We are given that u and v are a basis for the two dimensional vector space.
To prove that w=u+v and x=u-v is also a basis .
By using matrix we prove w and x are basis of vector space.
We make a matrix from w and x
[tex]\left[\begin{array}{cc}1&1\\1&-1\end{array}\right][/tex]
Apply operation
[tex] R_1\rightarrow R_1-R_2[/tex]
[tex]\left[\begin{array}{cc}0&2\\1&-1\end{array}\right][/tex]
Apply [tex] R_2\rightarrow R_2-+R_1[/tex]
[tex]\left[\begin{array}{cc}0&2\\1&1\end{array}\right][/tex]
Apply [tex]R_1\rightarrow \frac{1}{2}R_1[/tex]
[tex]\left[\begin{array}{cc}0&1\\1&1\end{array}\right][/tex]
Apply [tex]R_2\rightarrow R_2-R_1[/tex]
[tex]\left[\begin{array}{cc}0&1\\1&0\end{array}\right][/tex]
Rank is 2 .Therefore, row one and second row are linearly independent.
Hence, first and second row are linearly independent because, any row is not a linear combination of other row.
Therefore, w and x are formed basis of given vector space.
