Each condition required on $f$ can be turned into an equation involving the constants $a, b, c$.

In particular, $f(1) = 3$ tells us that $a + b + c = 3$.

Because $f'(x) = 2ax + b$, the condition $f'(1) = 3$ gives us $2a + b = 3$.

And finally $f^{\prime\prime}(x) = 2a$, and so $f^{\prime\prime}(1) = 2a = 2$. Thus we have the system of equations
\begin{align*}
a + b + c &= 3 \\
2a + b &= 3\\
2a &= 2
\end{align*}

To solve this system, we could create the augmented matrix and then reduce it.

For this system, though, it is simpler to solve directly. The third equation tells us that $a=1$.

Plugging this value into the second equation, we find $b=1$.

Plugging both of these values into the first equation, we see $c=1$ as well.

Thus the function we want is $f(x) = x^2 + x + 1$.

Find a Polynomial Satisfying the Given Conditions on Derivatives
Find a cubic polynomial
\[p(x)=a+bx+cx^2+dx^3\]
such that $p(1)=1, p'(1)=5, p(-1)=3$, and $ p'(-1)=1$.
Solution.
By differentiating $p(x)$, we obtain
\[p'(x)=b+2cx+3dx^2.\]
Thus the given conditions are
[…]

Is the Derivative Linear Transformation Diagonalizable?
Let $\mathrm{P}_2$ denote the vector space of polynomials of degree $2$ or less, and let $T : \mathrm{P}_2 \rightarrow \mathrm{P}_2$ be the derivative linear transformation, defined by
\[ T( ax^2 + bx + c ) = 2ax + b . \]
Is $T$ diagonalizable? If so, find a diagonal matrix which […]

12 Examples of Subsets that Are Not Subspaces of Vector Spaces
Each of the following sets are not a subspace of the specified vector space. For each set, give a reason why it is not a subspace.
(1) \[S_1=\left \{\, \begin{bmatrix}
x_1 \\
x_2 \\
x_3
\end{bmatrix} \in \R^3 \quad \middle | \quad x_1\geq 0 \,\right \}\]
in […]

Differentiation is a Linear Transformation
Let $P_3$ be the vector space of polynomials of degree $3$ or less with real coefficients.
(a) Prove that the differentiation is a linear transformation. That is, prove that the map $T:P_3 \to P_3$ defined by
\[T\left(\, f(x) \,\right)=\frac{d}{dx} f(x)\]
for any $f(x)\in […]

Exponential Functions Form a Basis of a Vector Space
Let $C[-1, 1]$ be the vector space over $\R$ of all continuous functions defined on the interval $[-1, 1]$. Let
\[V:=\{f(x)\in C[-1,1] \mid f(x)=a e^x+b e^{2x}+c e^{3x}, a, b, c\in \R\}\]
be a subset in $C[-1, 1]$.
(a) Prove that $V$ is a subspace of $C[-1, 1]$.
(b) […]

Taking the Third Order Taylor Polynomial is a Linear Transformation
The space $C^{\infty} (\mathbb{R})$ is the vector space of real functions which are infinitely differentiable. Let $T : C^{\infty} (\mathbb{R}) \rightarrow \mathrm{P}_3$ be the map which takes $f \in C^{\infty}(\mathbb{R})$ to its third order Taylor polynomial, specifically defined […]

Solve a System by the Inverse Matrix and Compute $A^{2017}\mathbf{x}$
Let $A$ be the coefficient matrix of the system of linear equations
\begin{align*}
-x_1-2x_2&=1\\
2x_1+3x_2&=-1.
\end{align*}
(a) Solve the system by finding the inverse matrix $A^{-1}$.
(b) Let $\mathbf{x}=\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}$ be the solution […]

Differentiating Linear Transformation is Nilpotent
Let $P_n$ be the vector space of all polynomials with real coefficients of degree $n$ or less.
Consider the differentiation linear transformation $T: P_n\to P_n$ defined by
\[T\left(\, f(x) \,\right)=\frac{d}{dx}f(x).\]
(a) Consider the case $n=2$. Let $B=\{1, x, x^2\}$ be a […]