Question: Write down \(\mathrm{T}\) if the following statement is true and \(\mathrm{F}\) if the statement is false. (2 points each) Example: \(\underline{\mathrm{F}}\) The map \(T(x)=x^{2}\) is a linear map on \(\mathbb{R}\).

1. The transformation \(T(A)=A^{T}\) from \(\mathbb{R}^{4 \times 4}\) to \(\mathbb{R}^{4 \times 4}\) is an isomorphism.
   \(n \times m\) matrix \(A\) and \(n\) -vector \(\vec{b}\), there is at least one solution to
2. For any \(n\)
   \[A^{T} A \vec{x}=A^{T} \vec{b}\]
3. There exist two distinct two-dimensional subspaces \(W_{1}\) and \(W_{2}\) of \(P_{5}\) such that the union \(W_{1} \cup W_{2}\) is a subspace of \(P_{5}\) as well.
4. If \(A\) is an orthogonal matrix, then \(A^{2}\) must also be orthogonal.
5. If \(A\) is a symmetric \(7 \times 7\) matrix \(\left(A^{T}=A\right)\), then the determinant of \(A\) is positive.
6. The following matrix is an orthogonal matrix:
   \[\frac{1}{7}\left[\begin{array}{ccc} 2 & 6 & -3 \\ 6 & -3 & 2 \\ 3 & 2 & 6 \end{array}\right]\]
7. If \(A\) is symmetric and \(S\) is orthogonal, then \(S^{-1} A S\) is symmetric as well.
8. If \(A\) is not a square matrix, then at least one of \(\operatorname{det}\left(A^{T} A\right)\) and \(\operatorname{det}\left(A A^{T}\right)\) must be zero.
9. The set \(\left\{f \in P_{2}: \int_{0}^{1} f(x) d x=1\right\}\) is a subspace of \(P_{2}\).
10. Every two-dimensional subspace of \(\mathbb{R}^{2 \times 2}\) contains at least one invertible matrix.

Price: $2.99

Solution: The downloadable solution consists of 2 pages
Deliverable: Word Document