Problems in Mathematics
Problems by Topics
Gauss-Jordan Elimination
Inverse Matrix
Linear Transformation
Eigen Value
Cayley-Hamilton Theorem
Diagonalization
Exam Problems
Group Homomorphism
Sylow's Theorem
Module Theory
Ring Theory
LaTex/MathJax
Login/Join us
Solve later Problems
My Solved Problems
You solved 0 problems!!
Solved Problems / Solve later Problems
by Yu · Published 12/26/2017 · Last modified 01/23/2018
The Set of Vectors Perpendicular to a Given Vector is a Subspace
Problem 659
Fix the row vector $\mathbf{b} = \begin{bmatrix} -1 & 3 & -1 \end{bmatrix}$, and let $\R^3$ be the vector space of $3 \times 1$ column vectors. Define
\[W = \{ \mathbf{v} \in \R^3 \mid \mathbf{b} \mathbf{v} = 0 \}.\] Prove that $W$ is a vector subspace of $\R^3$.
Read solution
Click here if solved 20
Add to solve later
by Yu · Published 12/26/2017
Prove that the Center of Matrices is a Subspace
Let $V$ be the vector space of $n \times n$ matrices with real coefficients, and define
\[ W = \{ \mathbf{v} \in V \mid \mathbf{v} \mathbf{w} = \mathbf{w} \mathbf{v} \mbox{ for all } \mathbf{w} \in V \}.\] The set $W$ is called the center of $V$.
Prove that $W$ is a subspace of $V$.
If $M, P$ are Nonsingular, then Exists a Matrix $N$ such that $MN=P$
Suppose that $M, P$ are two $n \times n$ non-singular matrix. Prove that there is a matrix $N$ such that $MN = P$.
A Condition that a Vector is a Linear Combination of Columns Vectors of a Matrix
Suppose that an $n \times m$ matrix $M$ is composed of the column vectors $\mathbf{b}_1 , \cdots , \mathbf{b}_m$.
Prove that a vector $\mathbf{v} \in \R^n$ can be written as a linear combination of the column vectors if and only if there is a vector $\mathbf{x}$ which solves the equation $M \mathbf{x} = \mathbf{v}$.
Show that the Given 2 by 2 Matrix is Singular
Consider the matrix $M = \begin{bmatrix} 1 & 4 \\ 3 & 12 \end{bmatrix}$.
(a) Show that $M$ is singular.
(b) Find a non-zero vector $\mathbf{v}$ such that $M \mathbf{v} = \mathbf{0}$, where $\mathbf{0}$ is the $2$-dimensional zero vector.
Column Vectors of an Upper Triangular Matrix with Nonzero Diagonal Entries are Linearly Independent
Suppose $M$ is an $n \times n$ upper-triangular matrix.
If the diagonal entries of $M$ are all non-zero, then prove that the column vectors are linearly independent.
Does the conclusion hold if we do not assume that $M$ has non-zero diagonal entries?
Write a Vector as a Linear Combination of Three Vectors
Write the vector $\begin{bmatrix} 1 \\ 3 \\ -1 \end{bmatrix}$ as a linear combination of the vectors
\[\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix} , \, \begin{bmatrix} 2 \\ -2 \\ 1 \end{bmatrix} , \, \begin{bmatrix} 2 \\ 0 \\ 4 \end{bmatrix}.\]
Prove that any Set of Vectors Containing the Zero Vector is Linearly Dependent
Prove that any set of vectors which contains the zero vector is linearly dependent.
Determine Trigonometric Functions with Given Conditions
(a) Find a function
\[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 \theta)\] such that $g(0) = g(\pi/2) = g(\pi) = 0$, where $a, b, c$ are constants.
(b) Find real numbers $a, b, c$ such that the function
\[g(\theta) = a \cos(\theta) + b \cos(2 \theta) + c \cos(3 \theta)\] satisfies $g(0) = 3$, $g(\pi/2) = 1$, and $g(\pi) = -5$.
Find a Quadratic Function Satisfying Conditions on Derivatives
Find a quadratic function $f(x) = ax^2 + bx + c$ such that $f(1) = 3$, $f'(1) = 3$, and $f^{\prime\prime}(1) = 2$.
Here, $f'(x)$ and $f^{\prime\prime}(x)$ denote the first and second derivatives, respectively.
Determine a 2-Digit Number Satisfying Two Conditions
A 2-digit number has two properties: The digits sum to 11, and if the number is written with digits reversed, and subtracted from the original number, the result is 45.
Find the number.
Determine Whether Matrices are in Reduced Row Echelon Form, and Find Solutions of Systems
Determine whether the following augmented matrices are in reduced row echelon form, and calculate the solution sets of their associated systems of linear equations.
(a) $\left[\begin{array}{rrr|r} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & -3 \\ 0 & 0 & 1 & 6 \end{array} \right]$.
(b) $\left[\begin{array}{rrr|r} 1 & 0 & 3 & -4 \\ 0 & 1 & 2 & 0 \end{array} \right]$.
(c) $\left[\begin{array}{rr|r} 1 & 2 & 0 \\ 1 & 1 & -1 \end{array} \right]$.
If a Symmetric Matrix is in Reduced Row Echelon Form, then Is it Diagonal?
Recall that a matrix $A$ is symmetric if $A^\trans = A$, where $A^\trans$ is the transpose of $A$.
Is it true that if $A$ is a symmetric matrix and in reduced row echelon form, then $A$ is diagonal? If so, prove it.
Otherwise, provide a counterexample.
Find All 3 by 3 Reduced Row Echelon Form Matrices of Rank 1 and 2
(a) Find all $3 \times 3$ matrices which are in reduced row echelon form and have rank 1.
(b) Find all such matrices with rank 2.
Click here if solved 103
If a Matrix $A$ is Full Rank, then $\rref(A)$ is the Identity Matrix
Prove that if $A$ is an $n \times n$ matrix with rank $n$, then $\rref(A)$ is the identity matrix.
Here $\rref(A)$ is the matrix in reduced row echelon form that is row equivalent to the matrix $A$.
If Two Matrices Have the Same Rank, Are They Row-Equivalent?
If $A, B$ have the same rank, can we conclude that they are row-equivalent?
If so, then prove it. If not, then provide a counterexample.
Find a Row-Equivalent Matrix which is in Reduced Row Echelon Form and Determine the Rank
For each of the following matrices, find a row-equivalent matrix which is in reduced row echelon form. Then determine the rank of each matrix.
(a) $A = \begin{bmatrix} 1 & 3 \\ -2 & 2 \end{bmatrix}$.
(b) $B = \begin{bmatrix} 2 & 6 & -2 \\ 3 & -2 & 8 \end{bmatrix}$.
(c) $C = \begin{bmatrix} 2 & -2 & 4 \\ 4 & 1 & -2 \\ 6 & -1 & 2 \end{bmatrix}$.
(d) $D = \begin{bmatrix} -2 \\ 3 \\ 1 \end{bmatrix}$.
(e) $E = \begin{bmatrix} -2 & 3 & 1 \end{bmatrix}$.
Row Equivalence of Matrices is Transitive
If $A, B, C$ are three $m \times n$ matrices such that $A$ is row-equivalent to $B$ and $B$ is row-equivalent to $C$, then can we conclude that $A$ is row-equivalent to $C$?
Find all Column Vector $\mathbf{w}$ such that $\mathbf{v}\mathbf{w}=0$ for a Fixed Vector $\mathbf{v}$
Let $\mathbf{v} = \begin{bmatrix} 2 & -5 & -1 \end{bmatrix}$.
Find all $3 \times 1$ column vectors $\mathbf{w}$ such that $\mathbf{v} \mathbf{w} = 0$.
Prove that $\mathbf{v} \mathbf{v}^\trans$ is a Symmetric Matrix for any Vector $\mathbf{v}$
Let $\mathbf{v}$ be an $n \times 1$ column vector.
Prove that $\mathbf{v} \mathbf{v}^\trans$ is a symmetric matrix.
Page 6 of 38« First«...345678910...2030...»Last »
This website's goal is to encourage people to enjoy Mathematics!
This website is no longer maintained by Yu. ST is the new administrator.
Linear Algebra Problems by Topics
The list of linear algebra problems is available here.
Elementary Number Theory (1)
Field Theory (27)
Group Theory (126)
Linear Algebra (485)
Math-Magic (1)
Module Theory (13)
Ring theory (67)
MathJax
Mathematical equations are created by MathJax. See How to use MathJax in WordPress if you want to write a mathematical blog.
How to Prove Markov's Inequality and Chebyshev's Inequality
How to Use the Z-table to Compute Probabilities of Non-Standard Normal Distributions
Expected Value and Variance of Exponential Random Variable
Condition that a Function Be a Probability Density Function
Conditional Probability When the Sum of Two Geometric Random Variables Are Known
Is the Set of All Orthogonal Matrices a Vector Space?
Every Basis of a Subspace Has the Same Number of Vectors
If a Half of a Group are Elements of Order 2, then the Rest form an Abelian Normal Subgroup of Odd Order
True or False. The Intersection of Bases is a Basis of the Intersection of Subspaces
How to Diagonalize a Matrix. Step by Step Explanation.
Determine Whether Each Set is a Basis for $\R^3$
Diagonalize a 2 by 2 Matrix $A$ and Calculate the Power $A^{100}$
Rank of the Product of Matrices $AB$ is Less than or Equal to the Rank of $A$
Express a Vector as a Linear Combination of Other Vectors
Find the Inverse Matrix Using the Cayley-Hamilton Theorem
The Intersection of Two Subspaces is also a Subspace
Prove Vector Space Properties Using Vector Space Axioms
How to Find a Basis for the Nullspace, Row Space, and Range of a Matrix
How to Find a Formula of the Power of a Matrix
Site Map & Index
abelian group augmented matrix basis basis for a vector space characteristic polynomial commutative ring determinant determinant of a matrix diagonalization diagonal matrix eigenvalue eigenvector elementary row operations exam finite group group group homomorphism group theory homomorphism ideal inverse matrix invertible matrix kernel linear algebra linear combination linearly independent linear transformation matrix matrix representation nonsingular matrix normal subgroup null space Ohio State Ohio State.LA probability rank ring ring theory subgroup subspace symmetric matrix system of linear equations transpose vector vector space
Search More Problems
Membership Level Free
If you are a member, Login here.
Problems in Mathematics © 2021. All Rights Reserved.