Tagged: exam

Top 10 Popular Math Problems in 2016-2017

It’s been a year since I started this math blog!!

More than 500 problems were posted during a year (July 19th 2016-July 19th 2017).

I made a list of the 10 math problems on this blog that have the most views.

Can you solve all of them?


The level of difficulty among the top 10 problems.
【★★★】 Difficult (Final Exam Level)
【★★☆】 Standard(Midterm Exam Level)
【★☆☆】 Easy (Homework Level)
 

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Determine Whether There Exists a Nonsingular Matrix Satisfying $A^4=ABA^2+2A^3$

Problem 486

Determine whether there exists a nonsingular matrix $A$ if
\[A^4=ABA^2+2A^3,\] where $B$ is the following matrix.
\[B=\begin{bmatrix}
-1 & 1 & -1 \\
0 &-1 &0 \\
2 & 1 & -4
\end{bmatrix}.\]

If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$.

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

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Compute $A^{10}\mathbf{v}$ Using Eigenvalues and Eigenvectors of the Matrix $A$

Problem 485

Let
\[A=\begin{bmatrix}
1 & -14 & 4 \\
-1 &6 &-2 \\
-2 & 24 & -7
\end{bmatrix} \quad \text{ and }\quad \mathbf{v}=\begin{bmatrix}
4 \\
-1 \\
-7
\end{bmatrix}.\] Find $A^{10}\mathbf{v}$.

You may use the following information without proving it.
The eigenvalues of $A$ are $-1, 0, 1$. The eigenspaces are given by
\[E_{-1}=\Span\left\{\, \begin{bmatrix}
3 \\
-1 \\
-5
\end{bmatrix} \,\right\}, \quad E_{0}=\Span\left\{\, \begin{bmatrix}
-2 \\
1 \\
4
\end{bmatrix} \,\right\}, \quad E_{1}=\Span\left\{\, \begin{bmatrix}
-4 \\
2 \\
7
\end{bmatrix} \,\right\}.\]

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

 

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Find a Basis of the Vector Space of Polynomials of Degree 2 or Less Among Given Polynomials

Problem 481

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)

 

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Determine Wether Given Subsets in $\R^4$ are Subspaces or Not

Problem 480

(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)
 

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Find an Orthonormal Basis of the Range of a Linear Transformation

Problem 478

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)

 

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Use the Cayley-Hamilton Theorem to Compute the Power $A^{100}$

Problem 471

Let $A$ be a $3\times 3$ real orthogonal matrix with $\det(A)=1$.

(a) If $\frac{-1+\sqrt{3}i}{2}$ is one of the eigenvalues of $A$, then find the all the eigenvalues of $A$.

(b) Let
\[A^{100}=aA^2+bA+cI,\] where $I$ is the $3\times 3$ identity matrix.
Using the Cayley-Hamilton theorem, determine $a, b, c$.

(Kyushu University, Linear Algebra Exam Problem)
 

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A Matrix Equation of a Symmetric Matrix and the Limit of its Solution

Problem 457

Let $A$ be a real symmetric $n\times n$ matrix with $0$ as a simple eigenvalue (that is, the algebraic multiplicity of the eigenvalue $0$ is $1$), and let us fix a vector $\mathbf{v}\in \R^n$.

(a) Prove that for sufficiently small positive real $\epsilon$, the equation
\[A\mathbf{x}+\epsilon\mathbf{x}=\mathbf{v}\] has a unique solution $\mathbf{x}=\mathbf{x}(\epsilon) \in \R^n$.

(b) Evaluate
\[\lim_{\epsilon \to 0^+} \epsilon \mathbf{x}(\epsilon)\] in terms of $\mathbf{v}$, the eigenvectors of $A$, and the inner product $\langle\, ,\,\rangle$ on $\R^n$.

 
(University of California, Berkeley, Linear Algebra Qualifying Exam)


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Null Space, Nullity, Range, Rank of a Projection Linear Transformation

Problem 450

Let $\mathbf{u}=\begin{bmatrix}
1 \\
1 \\
0
\end{bmatrix}$ and $T:\R^3 \to \R^3$ be the linear transformation
\[T(\mathbf{x})=\proj_{\mathbf{u}}\mathbf{x}=\left(\, \frac{\mathbf{u}\cdot \mathbf{x}}{\mathbf{u}\cdot \mathbf{u}} \,\right)\mathbf{u}.\]

(a) Calculate the null space $\calN(T)$, a basis for $\calN(T)$ and nullity of $T$.

(b) Only by using part (a) and no other calculations, find $\det(A)$, where $A$ is the matrix representation of $T$ with respect to the standard basis of $\R^3$.

(c) Calculate the range $\calR(T)$, a basis for $\calR(T)$ and the rank of $T$.

(d) Calculate the matrix $A$ representing $T$ with respect to the standard basis for $\R^3$.

(e) Let
\[B=\left\{\, \begin{bmatrix}
1 \\
0 \\
0
\end{bmatrix}, \begin{bmatrix}
-1 \\
1 \\
0
\end{bmatrix}, \begin{bmatrix}
0 \\
-1 \\
1
\end{bmatrix} \,\right\}\] be a basis for $\R^3$.
Calculate the coordinates of $\begin{bmatrix}
x \\
y \\
z
\end{bmatrix}$ with respect to $B$.

(The Ohio State University, Linear Algebra Exam Problem)

 

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True of False Problems on Determinants and Invertible Matrices

Problem 438

Determine whether each of the following statements is True or False.

(a) If $A$ and $B$ are $n \times n$ matrices, and $P$ is an invertible $n \times n$ matrix such that $A=PBP^{-1}$, then $\det(A)=\det(B)$.

(b) If the characteristic polynomial of an $n \times n$ matrix $A$ is
\[p(\lambda)=(\lambda-1)^n+2,\] then $A$ is invertible.

(c) If $A^2$ is an invertible $n\times n$ matrix, then $A^3$ is also invertible.

(d) If $A$ is a $3\times 3$ matrix such that $\det(A)=7$, then $\det(2A^{\trans}A^{-1})=2$.

(e) If $\mathbf{v}$ is an eigenvector of an $n \times n$ matrix $A$ with corresponding eigenvalue $\lambda_1$, and if $\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_2$, then $\mathbf{v}+\mathbf{w}$ is an eigenvector of $A$ with corresponding eigenvalue $\lambda_1+\lambda_2$.

(Stanford University, Linear Algebra Exam Problem)
 

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Subspace Spanned By Cosine and Sine Functions

Problem 435

Let $\calF[0, 2\pi]$ be the vector space of all real valued functions defined on the interval $[0, 2\pi]$.
Define the map $f:\R^2 \to \calF[0, 2\pi]$ by
\[\left(\, f\left(\, \begin{bmatrix}
\alpha \\
\beta
\end{bmatrix} \,\right) \,\right)(x):=\alpha \cos x + \beta \sin x.\] We put
\[V:=\im f=\{\alpha \cos x + \beta \sin x \in \calF[0, 2\pi] \mid \alpha, \beta \in \R\}.\]

(a) Prove that the map $f$ is a linear transformation.

(b) Prove that the set $\{\cos x, \sin x\}$ is a basis of the vector space $V$.

(c) Prove that the kernel is trivial, that is, $\ker f=\{\mathbf{0}\}$.
(This yields an isomorphism of $\R^2$ and $V$.)

(d) Define a map $g:V \to V$ by
\[g(\alpha \cos x + \beta \sin x):=\frac{d}{dx}(\alpha \cos x+ \beta \sin x)=\beta \cos x -\alpha \sin x.\] Prove that the map $g$ is a linear transformation.

(e) Find the matrix representation of the linear transformation $g$ with respect to the basis $\{\cos x, \sin x\}$.

(Kyoto University, Linear Algebra exam problem)

 

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