91Ó°ÊÓ

Let \(A, B\) be two linearly independent vectors in \(\mathbf{R}^{n}\). What is the dimension of the space perpendicular to both \(A\) and \(B ?\)

Let \(A\) be a triangular matrix $$ \left(\begin{array}{cccc} a_{11} & a_{12} & \ldots & a_{1 n} \\ 0 & a_{22} & \ldots & a_{2 n} \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & a_{n n} \end{array}\right) $$ Assume that none of the diagonal elements is equal to 0. What is the rank of \(A\) ?

Find the dimension of the space of solutions of the following systems of equations. Also find a basis for this space of solutions. (a) \(2 x+y-z=0\) (b) \(x-y+z=0\) \(y+z=0\) (c) \(4 x+7 y-\pi z=0\) (d) \(x+y+z=0\) \(2 x-y+z=0 \quad x-y=0\) \(y+z=0\)

Let \(W\) be the subspace of \(\mathbf{C}^{3}\) generated by the vector \((1, i, 0)\). Find a basis of \(W \perp\) in \(C^{3}\) (with respect to the ordinary dot produet of vectors).

Write out in full in terms of coordinates the expression for \({ }^{\prime} X A Y\) when \(A\) is the following matrix, and \(X, Y\) are vectors of the corresponding dimension. (a) \(\left(\begin{array}{rr}2 & -3 \\ 4 & 1\end{array}\right)\) (b) \(\left(\begin{array}{rr}4 & 1 \\ -2 & 5\end{array}\right)\) (c) \(\left(\begin{array}{rr}-5 & 2 \\ \pi & 7\end{array}\right)\) (d) \(\left(\begin{array}{rrr}1 & 2 & -1 \\ -3 & 1 & 4 \\ 2 & 5 & -1\end{array}\right)\) (e) \(\left(\begin{array}{rrr}-4 & 2 & 1 \\ 3 & 1 & 1 \\ 2 & 5 & 7\end{array}\right)\) (f) \(\left(\begin{array}{rrr}-\frac{1}{2} & 2 & -5 \\ 1 & 2 & 4 \\ -1 & 0 & 3\end{array}\right)\)

Let \(V\) be a vector space of dimension \(n\) over the field \(K .\) Let \(\psi, \varphi\) be two non-zero functionals on \(V\). Assume that there is no element \(c \in K, c \neq 0\) such that \(\psi=c \varphi .\) Show that $$ (\text { Ker } \varphi) \cap(\operatorname{Ker} \psi) $$ has dimension \(n-2\).

Let \(A X=B\) be a system of linear equations, where \(A\) is an \(m \times n\) matrix, \(X\) is an \(n\) -vector, and \(B\) is an \(m\) -vector. Assume that there is one solution \(X=X_{0} .\) Show that every solution is of the form \(X_{0}+Y\), where \(Y\) is a solution of the homogeneous system \(A Y=O\), and conversely any vector of the form \(X_{0}+Y\) is a solution.

(a) Let \(V\) be the vector space of all \(n \times n\) matrices over \(\mathbf{R}\), and define the scalar product of two matrices \(A, B\) by $$ \langle A, B\rangle=\operatorname{tr}(A B) $$ where tr is the trace (sum of the diagonal elements). Show that this is a scalar product and that it is non-degenerate. (b) If \(A\) is a real symmetric matrix, show that \(\operatorname{tr}(A A) \geqq 0\), and \(\operatorname{tr}(A A)\) \(>0\) if \(A \neq 0 .\) Thus the trace defines a positive definite scalar product on the space of real symmetric matrices. (c) Let \(V\) be the vector space of real \(n \times n\) symmetrie matrices. What is \(\operatorname{dim} V ?\) What is the dimension of the subspace \(W\) consisting of those matrices \(A\) such that \(\operatorname{tr}(A)=0 ?\) What is the dimension of the orthogonal complement \(W^{\perp}\) relative to the positive definite sealar product of part (b)?