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Linear Algebra -- Spring 2013 -- final exam

Name:_________________________

10 problems, 10 points each

• Show enough work to justify your answers. Refer to theorems whenever necessary.
• One page per problem.
1. (a) Prove that the set of all diagonal $n \times n$ matrices form a vector space. (b) A matrix is called upper-triangular if all entries below the main diagonal are equal to zero ($a_{ij}=0$ for $i>j$). Prove that the set of all upper-triangular $n\times n$ matrices form a vector space.
2. Prove that the intersection of two subspaces is always a subspace.
3. Give the definition of linear independence in vector spaces. Give an example of three linearly independent continuous functions.
4. Are the following operators from the vector space of all differential functions to itself linear? (a) $Q(f)=f$, (b) $T(f)=f+2$, (c) $D(f)=f'$, (d) $K(f)=|f|$, (e) $G(f)=3f$. Just the answers.
5. Find the inverse the matrix: \begin{equation*}A=\left[\begin{array}{cc}1 & 1 \\ 1 & 1\end{array}\right].\end{equation*}
6. Find the eigenvalues, the eigenvectors, and bases of the eigenspaces of the matrix: \begin{equation*}A=\left[\begin{array}{cc}1 & 0 \\ 1 & 0\end{array}\right].\end{equation*}
7. (a) Give the definition of an inner product space. (b) Give the definition of a normed space. (c) What is their relation?
8. Suppose $a$ is a vector in an inner product space $V$ and suppose $S$ is the set of all vectors orthogonal to $a$, plus $0$. Prove that $S$ is a subspace of $V$.
9. Find scalars $a$ and $b$ such that $a(1,2)+b(-1,3)=(1,12)$.
10. (a) What is the dimension of the space of all $3\times 3$ matrices? (b) What is the dimension of the space of all symmetric (i.e., $a_{ij}=a_{ji}$) $3\times 3$ matrices? (c) What is the dimension of the space of all upper-triangular $3\times 3$ matrices? Just the answers.
• Extra credit (10 points). Suppose $V$ is a vector space. Define: $V^*=\{f:V\rightarrow {\bf R}, linear \}.$ (a) Give $V^*$ a vector space structure. (b) Find $\dim \Big( {\bf R}^n \Big) ^*$.