91Ó°ÊÓ

Perform a rotation of axes to eliminate the xy-term, and sketch the graph of the conic. $$ 5 x^{2}-6 x y+5 y^{2}-12=0 $$

Find the rank of the matrix. \(\left[\begin{array}{rccccc}1 & 2 & 3 & . & . & n \\ n+1 & . n+2 & . n+3 & . . . & 2 n \\ 2 n+1 & 2 n+2 & 2 n+3 & . . . & 3 n \\ \vdots & \vdots & . & \vdots & & \vdots \\ n^{2}-n+1 & n^{2}-n+2 & n^{2}-n+3 & . . & n^{2}\end{array}\right]\)

When the set of vectors \(\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}, \ldots, \mathbf{u}_{n}\right\\}\) is linearly independent and the set \(\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}, \ldots, \mathbf{u}_{n}, \mathbf{v}\right\\}\) is linearly dependent, prove that \(\mathbf{v}\) is a linear combination of the \(\mathbf{u}_{i}^{\prime}\) s.

Let \(\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right\\}\) be a linearly independent set of vectors in a vector space \(V\). Delete the vector \(\mathbf{v}_{k}\) from this set and prove that the set \(\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}, \dots, \mathbf{v}_{k-1}\right\\}\) cannot span \(V\).

Proof Prove each property of the system of linear equations in \(n\) variables \(A \mathbf{x}=\mathbf{b}\) (a) If rank(A) \(=\operatorname{rank}([A, \mathbf{b}])=n,\) then the system has a unique solution. (b) If rank(A) \(=\operatorname{rank}([A \quad \mathbf{b}])

Perform a rotation of axes to eliminate the xy-term, and sketch the graph of the conic. $$ 7 x^{2}-6 \sqrt{3} x y+13 y^{2}-64=0 $$

Perform a rotation of axes to eliminate the xy-term, and sketch the graph of the conic. $$ 7 x^{2}-2 \sqrt{3} x y+5 y^{2}=16 $$

When \(V\) is spanned by \(\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{k}\right\\}\) and one of these vectors can be written as a linear combination of the other \(k-1\) vectors, prove that the span of these \(k-1\) vectors is also \(V\).

Let \(A\) be an \(m \times n\) matrix. Prove that \(N(A) \subset N\left(A^{T} A\right)\).

Proof Let \(A\) be an \(m \times n\) matrix. Prove that \(N(A) \subset N\left(A^{T} A\right)\)