Tagged: subspace
by Yu · Published 03/19/2018
Find a Spanning Set for the Vector Space of Skew-Symmetric Matrices
Let $W$ be the set of $3\times 3$ skew-symmetric matrices. Show that $W$ is a subspace of the vector space $V$ of all $3\times 3$ matrices. Then, exhibit a spanning set for $W$.
In which $\R^k$, are the Nullspace and Range Subspaces?
Suppose that the nullspace of $A$ is a plane in $\R^3$ and the range is spanned by a nonzero vector $\mathbf{v}$ in $\R^5$. Determine $m$ and $n$. Also, find the rank and nullity of $A$.
Find a basis for $\Span(S)$, where $S$ is a Set of Four Vectors
Find a basis for $\Span(S)$ where $S=
\left\{
\begin{bmatrix}
1 \\ 2 \\ 1
\end{bmatrix}
-1 \\ -2 \\ -1
2 \\ 6 \\ -2
\right\}$.
Find a Basis for the Subspace spanned by Five Vectors
Let $S=\{\mathbf{v}_{1},\mathbf{v}_{2},\mathbf{v}_{3},\mathbf{v}_{4},\mathbf{v}_{5}\}$ where
\mathbf{v}_{1}=
1 \\ 2 \\ 2 \\ -1
,\;\mathbf{v}_{2}=
1 \\ 3 \\ 1 \\ 1
1 \\ 5 \\ -1 \\ 5
.\] Find a basis for the span $\Span(S)$.
Can We Reduce the Number of Vectors in a Spanning Set?
Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^3$. Is it possible that $S_2=\{\mathbf{v}_1\}$ is a spanning set for $V$?
Does an Extra Vector Change the Span?
Suppose that a set of vectors $S_1=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\}$ is a spanning set of a subspace $V$ in $\R^5$. If $\mathbf{v}_4$ is another vector in $V$, then is the set
\[S_2=\{\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3, \mathbf{v}_4\}\] still a spanning set for $V$? If so, prove it. Otherwise, give a counterexample.
Describe the Range of the Matrix Using the Definition of the Range
Using the definition of the range of a matrix, describe the range of the matrix
\[A=\begin{bmatrix}
2 & 4 & 1 & -5 \\
1 &2 & 1 & -2 \\
1 & 2 & 0 & -3
\end{bmatrix}.\]
For Fixed Matrices $R, S$, the Matrices $RAS$ form a Subspace
Let $V$ be the vector space of $k \times k$ matrices. Then for fixed matrices $R, S \in V$, define the subset $W = \{ R A S \mid A \in V \}$.
Prove that $W$ is a vector subspace of $V$.
A Line is a Subspace if and only if its $y$-Intercept is Zero
Let $\R^2$ be the $x$-$y$-plane. Then $\R^2$ is a vector space. A line $\ell \subset \mathbb{R}^2$ with slope $m$ and $y$-intercept $b$ is defined by
\[ \ell = \{ (x, y) \in \mathbb{R}^2 \mid y = mx + b \} .\]
Prove that $\ell$ is a subspace of $\mathbb{R}^2$ if and only if $b = 0$.
Determine the Values of $a$ so that $W_a$ is a Subspace
For what real values of $a$ is the set
\[W_a = \{ f \in C(\mathbb{R}) \mid f(0) = a \}\] a subspace of the vector space $C(\mathbb{R})$ of all real-valued functions?
The Set $ \{ a + b \cos(x) + c \cos(2x) \mid a, b, c \in \mathbb{R} \}$ is a Subspace in $C(\R)$
Let $C(\mathbb{R})$ be the vector space of real-valued functions on $\mathbb{R}$.
Consider the set of functions $W = \{ f(x) = a + b \cos(x) + c \cos(2x) \mid a, b, c \in \mathbb{R} \}$.
Prove that $W$ is a vector subspace of $C(\mathbb{R})$.
The Centralizer of a Matrix is a Subspace
Let $V$ be the vector space of $n \times n$ matrices, and $M \in V$ a fixed matrix. Define
\[W = \{ A \in V \mid AM = MA \}.\] The set $W$ here is called the centralizer of $M$ in $V$.
Prove that $W$ is a subspace of $V$.
The Set of Vectors Perpendicular to a Given Vector is a Subspace
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 $\R^3$.
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 $V$.
Subspace Spanned by Trigonometric Functions $\sin^2(x)$ and $\cos^2(x)$
Let $C[-2\pi, 2\pi]$ be the vector space of all real-valued continuous functions defined on the interval $[-2\pi, 2\pi]$.
Consider the subspace $W=\Span\{\sin^2(x), \cos^2(x)\}$ spanned by functions $\sin^2(x)$ and $\cos^2(x)$.
(a) Prove that the set $B=\{\sin^2(x), \cos^2(x)\}$ is a basis for $W$.
(b) Prove that the set $\{\sin^2(x)-\cos^2(x), 1\}$ is a basis for $W$.
Is the Set of All Orthogonal Matrices a Vector Space?
An $n\times n$ matrix $A$ is called orthogonal if $A^{\trans}A=I$.
Let $V$ be the vector space of all real $2\times 2$ matrices.
Consider the subset
\[W:=\{A\in V \mid \text{$A$ is an orthogonal matrix}\}.\] Prove or disprove that $W$ is a subspace of $V$.
Find Bases for the Null Space, Range, and the Row Space of a $5\times 4$ Matrix
1 & -1 & 0 & 0 \\
0 &1 & 1 & 1 \\
0 & 2 & 2 & 2\\
0 & 0 & 0 & 0
(a) Find a basis for the null space $\calN(A)$.
(b) Find a basis of the range $\calR(A)$.
(c) Find a basis of the row space for $A$.
(The Ohio State University, Linear Algebra Midterm)
Find an Orthonormal Basis of the Given Two Dimensional Vector Space
Let $W$ be a subspace of $\R^4$ with a basis
\[\left\{\, \begin{bmatrix}
\end{bmatrix}, \begin{bmatrix}
\end{bmatrix} \,\right\}.\]
Find an orthonormal basis of $W$.
Vector Space of 2 by 2 Traceless Matrices
Let $V$ be the vector space of all $2\times 2$ matrices whose entries are real numbers.
\[W=\left\{\, A\in V \quad \middle | \quad A=\begin{bmatrix}
a & b\\
c& -a
\end{bmatrix} \text{ for any } a, b, c\in \R \,\right\}.\]
(a) Show that $W$ is a subspace of $V$.
(b) Find a basis of $W$.
(c) Find the dimension of $W$.
Find an Orthonormal Basis of $\R^3$ Containing a Given Vector
Let $\mathbf{v}_1=\begin{bmatrix}
2/3 \\ 2/3 \\ 1/3
\end{bmatrix}$ be a vector in $\R^3$.
Find an orthonormal basis for $\R^3$ containing the vector $\mathbf{v}_1$.
Symmetric Matrix and Its Eigenvalues, Eigenspaces, and Eigenspaces
Quiz 8. Determine Subsets are Subspaces: Functions Taking Integer Values / Set of Skew-Symmetric Matrices
Every Maximal Ideal of a Commutative Ring is a Prime Ideal
Find all Values of x such that the Given Matrix is Invertible
Torsion Subgroup of an Abelian Group, Quotient is a Torsion-Free Abelian Group