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# Linear Algebra -- Spring 2013 -- final exam

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10 problems, 10 points each

- Show enough work to justify your answers. Refer to theorems whenever necessary.
- One page per problem.

- (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.
- Prove that the intersection of two subspaces is always a subspace.
- Give the definition of linear independence in vector spaces. Give an example of three linearly independent continuous functions.
- 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.
- Find the inverse the matrix: \begin{equation*}A=\left[\begin{array}{cc}1 & 1 \\ 1 & 1\end{array}\right].\end{equation*}
- 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*}
- (a) Give the definition of an inner product space. (b) Give the definition of a normed space. (c) What is their relation?
- 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 $V$.
- Find scalars $a$ and $b$ such that $a(1,2)+b(-1,3)=(1,12)$.
- (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 $\dim \Big( {\bf R}^n \Big) ^*$.