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Exercises 3.5 Exercises
Exercise Group. In the following exercises find a basis and the dimension of vector space.
1. \(\C\) over \(\R\text{.}\) 2. \(F^n\) over a field \(F\) (refer to Example 2.2.2 )3. \(\End_{\R}(\R)\) over \(\R\) (refer to Example 2.2.3 ).4. \(M_{m\times n}(F)\) over a field \(F\text{.}\) 5. \(n\) over a field \(F\text{.}\)
Exercise Group. In the following exercises assume that \(V\) is a finite-dimensional vector space over a field \(F\text{.}\)
6. \(\{v_1,v_2,\ldots,v_n\}\) be a basis of \(V\text{.}\) Show that vectors
\begin{equation*}
w_i=\sum_{j=1}^{n}\alpha_{ij}v_j \quad \text{for } 1\leq i\leq m
\end{equation*}
are linearly dependent if and only if
\begin{equation*}
\begin{pmatrix}\alpha_{11}\amp\alpha_{21}\amp\cdots\amp\alpha_{m1}\\
\alpha_{12}\amp\alpha_{22}\amp\cdots\amp\alpha_{m2}\\
\vdots\amp\vdots\amp\ddots\amp\vdots\\
\alpha_{1n}\amp\alpha_{2n}\amp\cdots\amp\alpha_{mn}\\\end{pmatrix}X=0
\end{equation*}
has a nontrivial solution.7. \(\R\) considered as a vector space over \(\Q\) is not finite-dimensional.8. \(W<V\) is a proper subspace then, \(\dim_FW <\dim_FV\text{.}\) 9. \(1\) -dimensional subspaces of \(\R^2\text{.}\) 10. \(p\) a prime, and let \(\mathbb{F}_p\) be the field with \(p\) elements. Show the cardinality of a finite-dimensional vector space over \(\mathbb{F}_p\) is finite.11. \(V\) be a vector space over a field \(F\) with a basis \(\{v_1,v_2,\ldots,v_n\}\text{.}\) Show that every \(v\in V\) can be written uniquely as a linear combination of \(v_1,v_2,\ldots,v_n\text{.}\)
