Q. 1 : Let \( V \) be a vectior spater of finite dimension and let \( S=\left\{u_{1}, u_{2}, \ldots, u_{4}\right\} \) he a set iof liaculy independent vertors in \( V \). Then \( S \) is pat of a masis of \( V \); that is. \( S \) mus be exterterly of hasis of 1 .
Q. 2 : Suppose \( V \) has limite dimension and \( F: V \rightarrow U \) is linear. Then
10
\[
\operatorname{dim} V=\operatorname{dim}(\operatorname{ker} F)+\operatorname{dim}(\operatorname{im} F)=\operatorname{nullity}(F)+\operatorname{rank}(F)
\]
\( \alpha \) Let \( F: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} \) be detincal as \( \left.F(x, 1), z\right)=(x+3 y, 3 x-2 z, x-4 y-3 z) \) and comsialer the followang lmases of \( \mathbb{R}^{3} \) :
\[
E-\left\{c_{1}, \epsilon_{2}\right\}=\{(1,0,0) \text {. }
\]
\[
\left\{\mu_{1}, \|_{2}\right\} ?^{(6)}\{(1,0,0) \cdot(2,5,0) \cdot(0,5,2)\}
\]
Then
(i) show that \( f \) is a linear mapping.
(ii) Find the change-of-hasis matrix \( P \) from \( E \) to \( S \) and the change-of-basis matrix \( Q \) froun s back to \( E \).
(ii) Find the matrix \( A \) that represents \( F \) in the hasis \( E \).
(iv) Fiml the matrix \( B \) that represent \( F \) in the hasis \( S \).
Q. \( 4: L+1 I^{\circ}=M K_{n \times n}(K) \) and consider the following subspaces of \( V \) :
20
\( U= \) set of all syminetric matrices and \( W= \) set of all skew-symmetric matrices Then find a hasis of \( U \cap W \) and \( U+W \).
10)
