# Recent questions tagged vector

Let $L: V \rightarrow V$ be a linear map on a vector space $V$ and $z \in V$ a vector with the property that $L^{k-1} z \neq 0$ but $L^{k} z=0$. Show that $z, L z, \ldots L^{k-1} z$ are linearly independent.Let $L: V \rightarrow V$ be a linear map on a vector space $V$ and $z \in V$ a vector with the property that $L^{k-1} z \neq 0$ but $L^{k} z= ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(L: V \rightarrow V$ be a linear map on a vector space $V$.
Let $L: V \rightarrow V$ be a linear map on a vector space $V$.Let $L: V \rightarrow V$ be a linear map on a vector space $V$. a) Show that $\operatorname{ker} L \subset \operatorname{ker} L^{2}$ and, more g ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
For each of the sets $\mathcal{S}$ below, determine if it is a linear subspace of the given real vector space $V$. If it is a subspace, write down a basis for it.
For each of the sets $\mathcal{S}$ below, determine if it is a linear subspace of the given real vector space $V$. If it is a subspace, write down a basis for it.For each of the sets $\mathcal{S}$ below, determine if it is a linear subspace of the given real vector space $V$. If it is a subspace, write down ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Let $A$ be a real matrix, not necessarily square.
Let $A$ be a real matrix, not necessarily square.Let $A$ be a real matrix, not necessarily square. &nbsp; a) If $A$ is onto, show that $A^{}$ is one-to-one. b) If $A$ is one-to-one, show ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Give an example of a linear transformation $L: V \rightarrow V$ (or show that there is no such transformation) for which:
Give an example of a linear transformation $L: V \rightarrow V$ (or show that there is no such transformation) for which:Let $V$ be a vector space with $\operatorname{dim} V=10$ and let $L: V \rightarrow V$ be a linear transformation. Consider $L^{k}: V \rightarro ... close 0 answers 2 views If \(\mathbf{n}=(a, b, c)$ is a unit vector, use this formula to show that (perhaps surprisingly) the orthogonal projection of $\mathbf{x}$ into the plane perpendicular to $\mathbf{n}$ is given bya) Let $\mathbf{v}:=(a, b, c)$ and $\mathbf{x}:=(x, y, z)$ be any vectors in $\mathbb{R}^{3}$. Viewed as column vectors, find a $3 \times 3$ m ...
close
Given a unit vector $\mathbf{w} \in \mathbb{R}^{n}$, let $W=\operatorname{span}\{\mathbf{w}\}$ and consider the linear map $T: \mathbb{R}^{n} \rightarrow$ $\mathbb{R}^{n}$ defined byGiven a unit vector $\mathbf{w} \in \mathbb{R}^{n}$, let $W=\operatorname{span}\{\mathbf{w}\}$ and consider the linear map $T: \mathbb{R}^{n} \ri ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(L$ be a $2 \times 2$ matrix. For each of the following give a proof or counterexample.
Let $L$ be a $2 \times 2$ matrix. For each of the following give a proof or counterexample.Let $L$ be a $2 \times 2$ matrix. For each of the following give a proof or counterexample. a) If $L^{2}=0$ then $L=0$. b) If $L^{2}=L$ the ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Let $V, W$ be two-dimensional real vector spaces, and let $f_{1}, \ldots, f_{5}$ be linear transformations from $V$ to $W$.
Let $V, W$ be two-dimensional real vector spaces, and let $f_{1}, \ldots, f_{5}$ be linear transformations from $V$ to $W$.Let $V, W$ be two-dimensional real vector spaces, and let $f_{1}, \ldots, f_{5}$ be linear transformations from $V$ to $W$. Show that there ex ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Answer the following in terms of $\mathbf{V}, \mathbf{W}$, and $\mathbf{Z}$.
Answer the following in terms of $\mathbf{V}, \mathbf{W}$, and $\mathbf{Z}$.Let $A$ be a matrix, not necessarily square. Say $\mathbf{V}$ and $\mathbf{W}$ are particular solutions of the equations $A \mathbf{V}=\mathbf{ ... close 0 answers 2 views Let \(A: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ and $B: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$, so $B A: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3}$ and $A B: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2}$.Let $A: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2}$ and $B: \mathbb{R}^{2} \rightarrow \mathbb{R}^{3}$, so $B A: \mathbb{R}^{3} \rightarrow \mathb ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 For each of the following, answer TRUE or FALSE. If the statement is false in even a single instance, then the answer is FALSE. 0 answers 2 views For each of the following, answer TRUE or FALSE. If the statement is false in even a single instance, then the answer is FALSE.For each of the following, answer TRUE or FALSE. If the statement is false in even a single instance, then the answer is FALSE. There is no need to ju ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(V$ be a vector space and $\ell: V \rightarrow \mathbb{R}$ be a linear map.
Let $V$ be a vector space and $\ell: V \rightarrow \mathbb{R}$ be a linear map.Let $V$ be a vector space and $\ell: V \rightarrow \mathbb{R}$ be a linear map. If $z \in V$ is not in the nullspace of $\ell$, show that ever ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Proof or counterexample. In these $L$ is a linear map from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$, so its representation will be as a $2 \times 2$ matrix.
Proof or counterexample. In these $L$ is a linear map from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$, so its representation will be as a $2 \times 2$ matrix.Proof or counterexample. In these $L$ is a linear map from $\mathbb{R}^{2}$ to $\mathbb{R}^{2}$, so its representation will be as a $2 \times 2 ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Consider the homogeneous linear system \(A x=0$ where $A=\left(\begin{array}{rrrr} 1 & 3 & 0 & 1 \\ 1 & 3 & -2 & -2 \\ 0 & 0 & 2 & 3 \end{array}\right) \text {. }$
Consider the homogeneous linear system $A x=0$ where $A=\left(\begin{array}{rrrr} 1 & 3 & 0 & 1 \\ 1 & 3 & -2 & -2 \\ 0 & 0 & 2 & 3 \end{array}\right) \text {. }$Consider the homogeneous linear system $A x=0$ where \ A=\left(\begin{array}{rrrr} 1 &amp; 3 &amp; 0 &amp; 1 \\ 1 &amp; 3 &amp; -2 &amp; -2 \\ 0 &a ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Let $A$ be a square matrix with integer elements. For each of the following give a proof or counterexample.
Let $A$ be a square matrix with integer elements. For each of the following give a proof or counterexample.Let $A$ be a square matrix with integer elements. For each of the following give a proof or counterexample. a) If $\operatorname{det}(A)=\pm 1$, t ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Let $A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a real matrix, not necessarily square.
Let $A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a real matrix, not necessarily square.Let $A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a real matrix, not necessarily square. a) If two columns of $A$ are the same, show that $A ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a real matrix, not necessarily square.
Let $A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a real matrix, not necessarily square.Let $A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}$ be a real matrix, not necessarily square. a) If two rows of $A$ are the same, show that $A$ ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Let $A$ be a $4 \times 4$ matrix with determinant 7 . Give a proof or counterexample for each of the following.
Let $A$ be a $4 \times 4$ matrix with determinant 7 . Give a proof or counterexample for each of the following.Let $A$ be a $4 \times 4$ matrix with determinant 7 . Give a proof or counterexample for each of the following. &nbsp; a) For some vector $\mat ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Call a subset \(S$ of a vector space $V$ a spanning set if $\operatorname{Span}(S)=V$. Suppose that $T: V \rightarrow W$ is a linear map of vector spaces.
Call a subset $S$ of a vector space $V$ a spanning set if $\operatorname{Span}(S)=V$. Suppose that $T: V \rightarrow W$ is a linear map of vector spaces.Call a subset $S$ of a vector space $V$ a spanning set if $\operatorname{Span}(S)=V$. Suppose that $T: V \rightarrow W$ is a linear map of vec ...
close

Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993
Does an 8-dimensional vector space contain linear subspaces $V_{1}, V_{2}, V_{3}$ with no common non-zero element, such that
Does an 8-dimensional vector space contain linear subspaces $V_{1}, V_{2}, V_{3}$ with no common non-zero element, such thatDoes an 8-dimensional vector space contain linear subspaces $V_{1}, V_{2}, V_{3}$ with no common non-zero element, such that &nbsp; a). $\operato ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 What is the logarithmic norm? 1 answer 4 views What is the logarithmic norm?What is the logarithmic norm? ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Use Head to Tail Method to represent the following vectors and finally draw Resultant vector. 0 answers 17 views Use Head to Tail Method to represent the following vectors and finally draw Resultant vector.F1=12N in the positive x direction. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;F2=10N in the positive x direction. &nbsp;&nbs ... close Notice: Undefined index: avatar in /home/customer/www/mathsgee.com/public_html/qa-theme/AVEN/qa-theme.php on line 993 Why is \(|c| ||y|| = ||cy||$ where $y$ is a vector and $c$ is a constant?
Why is $|c| ||y|| = ||cy||$ where $y$ is a vector and $c$ is a constant?Why is $|c| ||y|| = ||cy||$ where $y$ is a vector and $c$ is a constant? ...