Linear algebra: final

Grading

Final exam for Linear algebra: course

10 problems, 10 points each

Instructions:

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

1. Show that the set of all even functions $f:\mathbf{R}\rightarrow \mathbf{R}$ is a vector space. What about the set of all the odd functions?
2. Give the definition of linear independence in vector spaces. Give examples of (a) three $2\times 2$ matrices that are linearly independent, and (b) three functions that are linearly dependent but not multiples of each other.
3. Suppose $A$ and $B$ are two invertible matrices. Express $(AB)^{-1}$ in terms of $A^{-1}$ and $B^{-1}.$ Prove.
4. Are the following functions linear? (a) $f(x,y,z)=(0,0,0);$ (b) $g(x,y,z)=(x-y,y-z,z-x);$ (c) $h(x,y,z)=(1,1,1);$ (d) $k(x,y,z)=||(x,y,z)||.$ Just the answers.
5. Suppose $\{v_{1},...,v_{m}\}$ is a linearly independent subset of a vector space $V$ and suppose $A:V\rightarrow U$ is a linear one-to-one operator. Prove that $\{Av_{1},...,Av_{m}\}$ is a linearly independent subset of $U.$
6. (a) Give the definition of the determinant of an $n\times n$ matrix. (b) Find the determinant of an upper-triangular matrix (all entries below the main diagonal are $0$). (c) Is the determinant a linear operator? Prove or disprove.
7. We know that if $S$ is a basis of $V$ then every element of $V$ can be represented as a linear combination of the elements of $S.$ Prove that such a representation is unique.
8. Find the eigenvalues, the eigenvectors, and bases of the eigenspaces of the matrix: \begin{equation*}A=\left[\begin{array}{cc}1 & 1 \\ 0 & 0\end{array}\right].\end{equation*}
9. (a) Give the definition of an inner product space. (b) State the Cauchy-Schwarz inequality. (c) Define the angle between two vectors in an inner product space. Prove that it's well-defined.
10. Suppose $a$ is an element of 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 $V$.

• Extra credit (5 points). Find the determinant of the $n\times n$ matrix with entries $1,2,3,\ldots ,n^{2}$. Prove.