Let $P_2$ be the vector space of all polynomials with real coefficients of degree $2$ or less.
Let $S=\{p_1(x), p_2(x), p_3(x), p_4(x)\}$, where
\begin{align*}
p_1(x)&=-1+x+2x^2, \quad p_2(x)=x+3x^2\\
p_3(x)&=1+2x+8x^2, \quad p_4(x)=1+x+x^2.
\end{align*}

(a) Find a basis of $P_2$ among the vectors of $S$. (Explain why it is a basis of $P_2$.)

(b) Let $B’$ be the basis you obtained in part (a).
For each vector of $S$ which is not in $B’$, find the coordinate vector of it with respect to the basis $B’$.

(The Ohio State University, Linear Algebra Final Exam Problem)

(a) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix}
x \\
y \\
z \\
w
\end{bmatrix}$ satisfying
\[2x+4y+3z+7w+1=0.\]
Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a subspace.

(b) Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix}
x \\
y \\
z \\
w
\end{bmatrix}$ satisfying
\[2x+4y+3z+7w=0.\]
Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a subspace.

(These two problems look similar but note that the equations are different.)

(The Ohio State University, Linear Algebra Final Exam Problem)

Let $T:\R^2 \to \R^3$ be a linear transformation given by
\[T\left(\, \begin{bmatrix}
x_1 \\
x_2
\end{bmatrix} \,\right)
=
\begin{bmatrix}
x_1-x_2 \\
x_2 \\
x_1+ x_2
\end{bmatrix}.\]
Find an orthonormal basis of the range of $T$.

(The Ohio State University, Linear Algebra Final Exam Problem)

Let
\[A=\begin{bmatrix}
1 & 2 & 1 \\
-1 &4 &1 \\
2 & -4 & 0
\end{bmatrix}.\]
The matrix $A$ has an eigenvalue $2$.
Find a basis of the eigenspace $E_2$ corresponding to the eigenvalue $2$.

(The Ohio State University, Linear Algebra Final Exam Problem)

Let $T:\R^2 \to \R^2$ be a linear transformation and let $A$ be the matrix representation of $T$ with respect to the standard basis of $\R^2$.

Prove that the following two statements are equivalent.

(a) There are exactly two distinct lines $L_1, L_2$ in $\R^2$ passing through the origin that are mapped onto themselves:
\[T(L_1)=L_1 \text{ and } T(L_2)=L_2.\]

(b) The matrix $A$ has two distinct nonzero real eigenvalues.

Let $G$ be a finite group of order $p^n$, where $p$ is a prime number and $n$ is a positive integer.
Suppose that $H$ is a subgroup of $G$ with index $[G:P]=p$.
Then prove that $H$ is a normal subgroup of $G$.

(Michigan State University, Abstract Algebra Qualifying Exam)

Give an example of two groups $G$ and $H$ and a subgroup $K$ of the direct product $G\times H$ such that $K$ cannot be written as $K=G_1\times H_1$, where $G_1$ and $H_1$ are subgroups of $G$ and $H$, respectively.

Prove that the symmetric group $S_n$, $n\geq 3$ is a semi-direct product of the alternating group $A_n$ and the subgroup $\langle(1,2) \rangle$ generated by the element $(1,2)$.

Let $W=C^{\infty}(\R)$ be the vector space of all $C^{\infty}$ real-valued functions (smooth function, differentiable for all degrees of differentiation).
Let $V$ be the vector space of all linear transformations from $W$ to $W$.
The addition and the scalar multiplication of $V$ are given by those of linear transformations.

Let $T_1, T_2, T_3$ be the elements in $V$ defined by
\begin{align*}
T_1\left(\, f(x) \,\right)&=\frac{\mathrm{d}}{\mathrm{d}x}f(x)\\[6pt]
T_2\left(\, f(x) \,\right)&=\frac{\mathrm{d}^2}{\mathrm{d}x^2}f(x)\\[6pt]
T_3\left(\, f(x) \,\right)&=\int_{0}^x \! f(t)\,\mathrm{d}t.
\end{align*}
Then determine whether the set $\{T_1, T_2, T_3\}$ are linearly independent or linearly dependent.