For questions about vector spaces and their properties. More general questions about linear algebra belong under the linear-algebra tag. A vector space is a space which consists of elements called "vectors", which can be added and multiplied by scalars. In other words, these are the spaces where we ...
0
votes
2answers
41 views
Finite Dimensional Subspaces and Their Properties
Let $W_1$ and $W_2$ be finite dimensional subspace fo a vector space $V$. How should I start to prove that the subspaces $W_1 \cap W_2$ and $W_1+W_2$ are also finite dimensional and
\begin{eqnarray}
...
1
vote
1answer
22 views
Vector cross product
I have this question on my take home assignment and it is giving me a headache.
Find the volume of the parallelepiped with three edges formed by <2,1,0> , <-1,2,0> and <1,1,2> using the ...
0
votes
1answer
23 views
Two linear functionals having the same Kernel are proportional
Let $V$ be a $k$-vector space, of finite dimension. Let $F,G:V\longrightarrow K$ be two non-zero $k$-linear applications. Suppose that $F$ and $G$ have the same kernel. Then $F$ and $G$ are ...
1
vote
1answer
35 views
Orthogonal complement of S in a finite field $\mathbb F_q$
For the following set S and corresponding finite field $\mathbb F_q$, fing the $\mathbb F_q$-linear span $\left<S\right>$ and its orthogonal complement $\left<S\right>$-perp.
$$S = ...
3
votes
4answers
74 views
If $X$ is an orthogonal matrix, why does $X^TX = I$?
It's not immediately clear to me why this is true. My notes say that putting $n$ orthonormal vectors $ v_1, ..., v_n$ in the columns of $X$ gives $X^TX = I$, and it follows from this that the rows of ...
1
vote
1answer
104 views
Real and complex canonical forms of quadratic form
How do I find the canonical form of
$$q_1(x,y,z)= 4x^2 +4xz+2yz$$
Now I have put it in matrix form as:
$$\left(
\begin{matrix}
4 & 0 & 2 \\
0 & 0 & 1 \\
...
1
vote
2answers
31 views
Basis of complex matrix vector space over $\Bbb{R}$
I understand that the basis of the vector space $$Mat_2(\Bbb{R}) = \begin{pmatrix}a_{11} & a_{12} \\ a_{21} & a_{22}\end{pmatrix}$$ over $\Bbb{R}$ is $$e = \left\{ \begin{pmatrix}1 & 0\\ 0 ...
4
votes
1answer
46 views
Inner product spaces that are isometrically isomorphic
I know this is a fundamental result in linear algebra, and although it is referenced in my textbook, it does not have a proof for it. I was wondering if someone could help me out:
Let $V$ and $W$ be ...
4
votes
3answers
203 views
If $Ax=B$ has two solution, then there must be a third one?
How do I prove this conjecture?
Let $A$ be a matrix, and $B$ be a column vectore. If $Ax=B$ has two solutions, then there must be a third one.
Thanks in a advance!
1
vote
1answer
38 views
finding a suitable orthogonal matrix
Assume that $x,y \in \text{R}^n$ are two arbitrary vectors. Assuming that $x^\top y >0$, I want to prove that there exist an orthogonal matrix $U \in \text{O}(n)$ such that all elements of the ...
0
votes
1answer
51 views
Prove that D (the differential operator) maps V (a vector space) into V.
I'm quite confused about what "into" means here and, more importantly, how I am supposed to prove that something maps a vector space into (not onto) another vector space.
Here's some of the ...
1
vote
1answer
40 views
What is the normal form for this line?
I have calculated the parametric form of a line as: $L = P_1 + tP_1P_3 = <2,2,0> + t<1,2,2>$.
If I am given a point $ K = <1,-1,-1>$, how would I show the normal form of plane $E$ ...
2
votes
1answer
52 views
Proving linear independence
Let $A$ be an $n \times n$ matrix and suppose $v_1, v_2, v_3 \in \mathbb{R}^n$ are nonzero vectors that satisfy:
$$
Av_1 = v_1 \\
Av_2 = 2v_2 \\
Av_3 = 3v_3 $$
Prove that $\{v_1, v_2, v_3\}$ is ...
2
votes
2answers
51 views
Linear algebra prove or disprove Kernel and Range
For a linear map $h: \mathbb R^3 \rightarrow \mathbb R^2$, the kernel of $h$ is a subspace of $ \mathbb R^2$.
For a linear map $h: \mathbb R^3 \rightarrow \mathbb R^2$, the range of $h$ is a subspace ...
3
votes
5answers
220 views
Dimensions: $\bigcap^{k}_{i=1}V_i \neq \{0\}$
Let $V$ be a vector space of dimension $n$ and let $V_1,V_2,\ldots,V_k \subset V$ be subspaces of
$V$. Assume that
\begin{eqnarray}
\sum^{k}_{i=1} \dim(V_i) > n(k-1).
\end{eqnarray}
To show that ...
2
votes
3answers
39 views
Linear algebra - how to tell where vectors lie?
I'm working my way (self-study) through Strang's text on Linear Algebra and am currently on Problem 1.2 #6.
6b) The vectors that are perpendicular to $V = (1,1,1)$ lie on a _ .
6c) The vectors that ...
0
votes
1answer
41 views
Kernels of Adjoints
Let $A$ be an $m \times n$ matrix. Show that $\mbox{Ker} A = \mbox{Ker} (A^*A)$.
To do that you need to prove 2 inclusions, $\mbox{Ker} (A^*A)$ is a subset of $\mbox{Ker} A$ and $\mbox{Ker} A$ is a ...
2
votes
2answers
37 views
Generalised eigenvalue is eigenvalue if it is in the field
I would like to prove the following assertion:
Let $\mathscr{F}$ be a field and $\mathscr{\phi}$ be an $\mathscr{F}$-linear endomorphism of a finite dimensional $\mathscr{F}$-vector space ...
-2
votes
1answer
41 views
Determine Span of vectors?
I did not see question like this before?
What is span of $(1,1+x,1+x+x^2,....,1+x+x^2+...+x^n)$ ?
The question also says to let $V=P_n(X)$ be the space of all polynomials whose degrees are less than ...
0
votes
1answer
35 views
What is required to establish the law of cosines?
In my quantum computation course, we have been given nothing more than the basic axioms of a linear vector space, and and the properties of an inner product; but we have started referring to "the ...
0
votes
2answers
42 views
Counter Example: Span Inclusion
For any two subsets $S$ and $S'$ of a vector space $V$ does $span(S) \cap span(S') = span(S \cap S')$?
If $S=ax, a \in \mathbb{R}$ in $\mathbb{R}^2$ and $S'=by, b \in \mathbb{R}$ in $\mathbb{R}^2$ ...
5
votes
6answers
125 views
Show that every subspace of $\mathbb{R}^n$ is a kernel of a linear map.
Let $S$ be a subspace of $\mathbb R^n$. Show that there is an $n \times n$ matrix $A$ such that
$$S= \{x \in \mathbb R^n : Ax=0\}.$$
How to proceed?
2
votes
2answers
133 views
If a subset $S$ of a vector space $V$ is a subspace of $V$, then is $\langle S \rangle = S$?
I'm reading here on page 22 of Axler, Linear Algebra Done Right, where the following is stated:
A $\bf{linear}$ $\bf{combination}$ of a list $(v_1,\dots,v_m)$ of vectors in $V$ is a vector of the ...
0
votes
0answers
36 views
How to show that $\mathrm{Sym}_{n\times n}(\Bbb{R})$ and $\mathrm{Skew}_{n\times n}(\Bbb{R})$ are subspaces of $\mathrm{M}_{n\times n}(\Bbb{R})$
A matrix $M \in \mathrm{M}_{n\times n}(\mathbb{R})$ is called symmetric (respectively, skew-symmetric) if $M^t = M$ (respectively, $M^t = -M$). How does one prove that the sets $\mathrm{Sym}_{n\times ...
2
votes
1answer
60 views
Vector Spaces Linear algebra
bI've been working through some problems in my Linear Algebra course and I've come across some that have me confused. I'm not particularly good at vector spaces so some help would be greatly ...
-6
votes
2answers
127 views
Intersection | Subspaces | Span
If $W_1$ and $W_2$ are two subspaces of a vector space $V$, then $W_1+W_2$ is the intersection of all subspaces of $V$ that contain $W_1$ and $W_2$, right? Is the intersection of all subspaces of $V$ ...
1
vote
2answers
36 views
Picture of Space Layout
If $W_1+W_2$ is the intersection of all subspaces of $V$ that contain $W_1$ and $W_2$, then how should this be represented pictorially? Also, how do I prove that $W_1+W_2$ is the intersection of all ...
1
vote
2answers
39 views
Find an orthogonal vector to 2 vector
I have the following problem:
A B C D are the 4 consecutive summit of a parallelogram, and have the following coordinates
A(1,-1,1);B(3,0,2);C(2,3,4);D(0,2,3)
I must find a vector that is ...
1
vote
1answer
38 views
Question on finding an orthogonal complement
So I have a practice question, and I want to make sure that my understanding of the concept is valid. This is the text of the question:
Find the orthogonal complement of the subspace of $ R^3$ ...
1
vote
2answers
23 views
If $L=\{B : BA = 0 \}$ and $R=\{C : AC = 0 \}$, what is the dimension of $L$, and $R$? $L,R,\{A\} \subset \mathbb R_{n \times n}$
Let $A \in \mathcal{M}_{n \times n}$ be a square matrix of size $n$ with rank $k \in [0,n]$. Define
$$
L = \{B \in \mathcal{M}_{n \times n} : BA = 0 \}, \\
R = \{C \in \mathcal{M}_{n \times n} : AC = ...
0
votes
2answers
37 views
Is Null equal to $\mathbb{R}^n$
Say I have a matrix $A$ and its row reduced echelon form looks like this:
$$
\begin{bmatrix}
1 &-3 &2 &-7\\
0 &0 &0 &0\\
0 &0 &0 &0\\
0 &0 &0 &0
...
2
votes
2answers
37 views
If $\dim(A+B)=\dim(A\cap B)+1$, then $A \subset B$ or $B \subset A$
Let $A,B$ be linear subspaces of $\mathbb R^n$. Show that if
$$
\dim(A+B) = \dim(A \cap B) + 1, \tag{1}
$$
then one of the space is a subset of the other,
$$
A \subset B \text{ or } B \subset A. ...
1
vote
1answer
46 views
Why translation of vectors doesn't preserve the cosinus of the angle they form?
Why augmenting two vectors by adding a constant scalar at each dimensions of two vectors doesn't preserve their cosinus while multiplying with a scalar (scaling) does preserve it?The first operation ...
1
vote
1answer
26 views
Basis of one linear operator form the other linear operator.
Suppose that I have two linear operators, $T_1$ and $T_2$ represented by matrices $3x3$.
Suppose that I have found the eigenbasis of $T_1$ equals to:
$$\{ \begin{pmatrix}
1\\
0\\
-1
...
0
votes
1answer
34 views
Lin Alg- Dual Spaces
Let $(V^*)^*=V^{**}$. Define $S:V\to V^{**}$ by $s(v)(\alpha)=\alpha(v)$ for all $v\in V$ and $\alpha\in V^*$.
I need to show that $s(v)\in V^{**}$. And show the S is a linear transformation. ($V$ is ...
0
votes
1answer
32 views
Normalized/Unit vector
I am confused with the following -
Do unit vectors and normalized vectors indicate the same thing?
If I have a vector, and if I divide it by the length of the vector, then what will we get?
-3
votes
1answer
46 views
Problem # 3 linear algebra [closed]
Let $V$ be a finite-dimensional vector space and let $W_{1}$ be any subspace of $V$. Prove that there is a subspace $W_{2}$ of $V$ such that $V= W_{1}\oplus W_{2}$.
Theorems that can be used:
...
0
votes
1answer
38 views
Proving the range $R$ and null space $N$ of a linear operator are independent iff $V = R \oplus N$
Let $T$ be a linear operator on a finite-dimensional vector space $V$. Let $R$ be the range of $T$ and let $N$ be the null space of $T$. Prove that $R$ and $N$ are independent if and only if $V= R ...
3
votes
1answer
44 views
Dense subspaces
How does one go about proving the following statements?
(a) $\operatorname{Lip}[a,b]$ functions are dense in absolutely continuous functions on $[a,b]$ under the variation norm - (Another doubt: what ...
-1
votes
1answer
54 views
Problem 1 linear algebra* [closed]
Let $E$ be a projection of $V$ and $T$ be a linear operator on $V$. Prove that the range of $E$ is invariant under $T$ if and only if $ETE=TE$ . Prove that both the range and the null space of $E$ ...
3
votes
1answer
96 views
Let $n$ be an integer $\geq 2$ and let $M_n(\Bbb R)$ denote…
Let $n$ be an integer $\geq 2$ and let $M_n(\Bbb R)$ denote the vector space of $n \times n$ real matrices. Let $B \in M_n(\Bbb R)$be an orthogonal matrix and let $B^t$ denote the transpose of $B.$ ...
0
votes
1answer
73 views
Can anyone comprehend fourth-dimensional space and higher visually? [closed]
When I look at the "Clifford torus," for example, it just looks like a three dimensional object that's morphing/changing shape as it moves. Can anyone actually comprehend fourth-dimensional objects ...
0
votes
2answers
52 views
Linear Algebra: Why this is not a reduced row echelon form?
I know, the following $4\times 1$ matrices:
$$\begin{pmatrix}1\\0\\0\\0\end{pmatrix},\begin{pmatrix}0\\0\\0\\0\end{pmatrix}$$
are in reduced row echelon form, but why is:
...
3
votes
1answer
39 views
Effect of doubly stochastic matrix on vector norm
Let $D$ be a $N \times N$ doubly stochastic matrix, $x$ be a $N$ dimensional vector.
What is the relation between $\Vert Dx \Vert_2$ and $\Vert x \Vert_2$?
In addition if $\Vert x \Vert_2=1$, what ...
2
votes
1answer
25 views
Finding the basis for a plane
I know the conditions of being a basis. The vectors in set should be linearly independent and they should span the vector space.
However, how do I find two different sets of basis vectors of the ...
1
vote
0answers
22 views
A modular which is not a metrizing modular (hence not an F-norm)?
I'm taking the terminology from Rolewicz's 1985 Metric Linear Spaces.
Given a complex vector space $X$, a modular $m$ is any function $m:X\to[0,+\infty]$ satisfying the following for all $x,y\in X$ ...
1
vote
1answer
51 views
Solution set to cross product
If $\vec a,\vec b \in \mathbb{R}^3$ with $|\vec a|\ne0$ show that the equation $\vec a \times \vec u =\vec b$ has a solution if and only if $a \cdot b = 0$ and find all the solutions in this case.
...
1
vote
5answers
126 views
A linear transformation $T$ is one-to-one if and only if it is onto
Let $V$ be a finite dimensional vector space. Show that a linear transformation $T\colon V \to V$ is one-to-one if and only if it is onto.
The hint I was given was that one only needs to show that ...
0
votes
1answer
82 views
How to verify this is an orthogonal basis? How to transform it into an orthonormal basis?
Let
$$B = \left\{ \begin{bmatrix} 3\\ -3\\ 0\end{bmatrix},\begin{bmatrix} 2\\ 2\\ -1\end{bmatrix},\begin{bmatrix} 1\\ 1\\ 4\end{bmatrix}\right\},\qquad v =\begin{bmatrix} 5\\ -3\\ ...
-1
votes
2answers
64 views
Express each of the standard basis vectors as linear combination of $\alpha_1,\alpha_2,\alpha_3$
Given the vectors $\alpha_1=(1,0,-1)$, $\alpha_2=(1,2,1)$, $\alpha_3=(0,-3,2)$, express each of the standard basis vectors as linear combination of $\alpha_1$, $\alpha_2$, and $\alpha_3$.
