Question

Suppose u and v are a basis for the two-dimensional vector space V. Prove that w = u+v and x = u-v is also a basis. Hint: If two vectors form a basis, then any vector in the space can be expressed as linear combinations of the two vectors. You know that u and v are a basis. Pick any vector, call it s, in the space and check that you can always do the same using w and x.

Answer 1

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

$\left[\begin{array}{cc}1&1\\1&-1\end{array}\right]$

Apply operation

$R_1\rightarrow R_1-R_2$

$\left[\begin{array}{cc}0&2\\1&-1\end{array}\right]$

Apply $R_2\rightarrow R_2-+R_1$

$\left[\begin{array}{cc}0&2\\1&1\end{array}\right]$

Apply $R_1\rightarrow \frac{1}{2}R_1$

$\left[\begin{array}{cc}0&1\\1&1\end{array}\right]$

Apply $R_2\rightarrow R_2-R_1$

$\left[\begin{array}{cc}0&1\\1&0\end{array}\right]$

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.