For questions about vector spaces and their properties. More general questions about linear algebra belong under the [tag: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 ...

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2
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1answer
54 views

projective geometry and relationship of cross-ratios

Define for pairwise different points $P_i=[v_i]$ the cross-ratio $\operatorname{CR}(P_1,P_2,P_3,P_4) = \frac{\det(v_1,v_2)}{\det(v_2,v_3)}\cdot\frac{\det(v_3,v_4)}{\det(v_4,v_1)}$ on $\mathbb{KP^1}$ ...
0
votes
1answer
43 views

For ${\bf x} = (x_1, x_2) \in R$ find ${\bf y} = (y_1, y_2)$ such that ${\bf x} = y_1{\bf v_1} + y_2{\bf v_2}$

For ${\bf x} = (x_1, x_2) \in R$ find ${\bf y} = (y_1, y_2)$ such that ${\bf x} = y_1{\bf v_1} + y_2{\bf v_2}$ and find a matrix $M \in M_2(R)$ such that ${\bf y} = M{\bf x}$. How do I go about ...
0
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2answers
32 views

How do you rotate a vector by $90^{\circ}$?

Consider a vector $\vec{OA}$. How will I rotate this vector by $90^{\circ}$ and represent in algebraically?
-1
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1answer
46 views

Linear transformation and characteristic polynomial

Let $V $ be an $n$-dimensional vector space and $T : V \to V$ a non-invertible linear transformation. Show that there is a subspace $W \subset V$ which is $(n−1)$-dimensional and contains ...
1
vote
1answer
49 views

Vector Space Dimension

Let $A,B$ be $n\times m$, $s\times m$ matrices respectively, and let $$V=\{X\in \mathbb{F}^{m\times n};\ B X A=0\}.$$ Suppose that $$rank(A)=r,\ rank(B)=m.$$ Show that $dim V=m(n-r)$. I have no idea. ...
0
votes
1answer
14 views

A question regarding lines between points.

On pg.13 of Lang's "Second Course in Calculus", the following is asserted: Let $P=(2,1)$ and $A=(-1,5)$. Then the parametric equation of the line through $P$ and in the direction of $A$ gives us ...
1
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0answers
57 views

vector calculus work done line integral

A string lies along the circle $x^2 + y^2 = 4$ from $(2,0)$ to $(0,2)$ in the first quadrant. The density of the string is $ρ(x,y) = xy$. a. Partition the string into a finite number of sub-arcs to ...
1
vote
1answer
29 views

linear dependence on $\mathbb R^n$

$S_1$ and $S_2$ finite sets on $\mathbb{R}^n$, $S_1$ is a subset of $S_2$, $(S_1\neq S_2)$. If $S_2$ is linearly dependent, so: $S_1$ could be linearly dependent? $S_1$ Could be linearly ...
0
votes
1answer
93 views

bijective maps leaving cross-ratio invariant are just the projective transformations

Show the bijective correspondence between a. bijective maps $f: \mathbb{P(K^1)} \to \mathbb{P(K^1)}$ which keep the cross-ratio invariant b. projective transformationes, i.e. $f: \mathbb{P(K^1)} \to ...
1
vote
1answer
132 views

Axiom of choice and vector space bases

Is this: for every vector space $V$, if $B$ and $C$ are bases of $V$, then there is a bijection: $B\to C$ iff the axiom of choice holds true? Or, perhaps, if axiom of choice is replaced by ...
0
votes
1answer
71 views

Necessary and Sufficient Condition for Vector Space

Problem Assume a finite set $F$, write the necessary and sufficient condition in terms of the number of elements of $F$, such that $F$ is a real vector space. (Assuming that the vector addition and ...
2
votes
3answers
43 views

Prove that $\mathbb{R}_{\neq0}$ is not a real vector space

Assume a set of $\mathbb{R}_{\neq0}=\{a \in \mathbb{R} \mid a \neq 0\}$, where addition of elements in $\mathbb{R}_{\neq0}$ is the product in scalar $ab$. Prove that this is not a real vector space. ...
2
votes
2answers
170 views

Linear algebra question with dual basis?

I have $B= \{b_1, \dots,b_n \}$ be basis for vector space $V$ over reals. Then if $A = \{a_1, \dots,a_n \}$ be basis for dual space $V^*$ (dual space is defined as set of all linear function mapping ...
5
votes
3answers
423 views

Why a non-diagonalizable matrix can be approximated by an infinite sequence of diagonalizable matrices?

It is known that any non-diagonalizable matrix, $A$, can be approximated by a set of diagonalizable matrices, e.g. $A \simeq \lim_{k \rightarrow \infty} A_k$. Why this is true? Note: I was faced with ...
1
vote
0answers
145 views

Dual Space Annihilator Question

$T\colon V\to W$ is a linear map, with $V$ being finite dimensional. Show that $\mbox{Im }T'=(\ker T)^0$ where ' refers to dual space and $^0$ refers to the annihilator of a space. My confustion with ...
0
votes
1answer
75 views

If $Ker(T^r)=Ker(T^{r+n})$ does that imply $Im(T^r)=Im(T^{r+n})$

Here T is a linear transformation and an n dimensional vector space V, $T:V\rightarrow V$. And $r$ is such that $Ker(T^r)=Ker(T^{r+j})\ \forall j\geq1$ Also, how can I show that $W=T^r(V)$ is T- ...
4
votes
1answer
83 views

Has anyone succeeded in formalizing the notion of a complete vector space? (Not using topological ideas).

In order theory, we have the concept of a lattice, which is defined as consisting of an underlying set $L$ together with two binary operations $\wedge$ and $\vee$. Now when $L$ is finite, the concept ...
6
votes
1answer
98 views

Linear and Commutative function over Square Matrices.

Find all functions $f$, such that $f(mA+nB) = mf(A) + nf(B)$ and $f(AB) = f(BA)$ , where $A, B$ are square matrices and $ m,n$ are scalars. Need to find $f$ as an explicit function of any general ...
0
votes
1answer
48 views

Finding the change of a basis from 1 matrix to another

I know how to convert a given Basis B to a standard basis. I know how to find a coordinate vector with respect to B given the standard basis. But how do I find the change of basis from a Matrix to ...
2
votes
1answer
72 views

Euclidian Spaces

Let $T$ be a linear operator on $V=\mathbb{R}^n$ whose matrix $A$ is a real symmetric matrix. Could someone show me how to prove that $V=(\text{ker } T) \oplus (\text{Im } T)$ ?
1
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2answers
249 views

Orthogonal unit vectors

Let $W$ be a two-dimensional subspace of $\mathbb{R}^3$, and consider the orthogonal projection $\pi$ of $\mathbb{R}^3$ onto W. Let $(a_i,b_i)^t$ be the coordinate vector of $\pi(e_i)$, with respect ...
1
vote
1answer
118 views

Let V be a finite dimensional subspace

1)Let $T : V \to V$ be a nonzero linear operator. Prove that either $T$ is an isomorphism, or there exists nonzero $R$, $S$ in $L(V, V)$ such that $RT = O$ and $TS = O$. 2)Let $I(V)$ be the set of ...
0
votes
1answer
79 views

What is the null vector for the vector space of continuous functions $f \colon \mathbf{R} \to \mathbf{C}$?

The set of all continuous complex-valued functions of real variable $x$ together with addition of two vectors $\boldsymbol{f} = f(x)$ and $\boldsymbol{g} = g(x)$ defined by \begin{equation} ...
3
votes
2answers
228 views

Can you factor out vectors?

My prof introduced eigenvalues to us today: Let $A$ be an $n \times n$ matrix. If there a scalar $\lambda$ and an $n-1$ non-zero column vector $u$, then $$Au = \lambda u$$ then $\lambda$ ...
1
vote
1answer
40 views

Basis for vector space given combination of vector components

The following is the first step in a homework problem of mine: Find a basis for the vector space $S = \{(x,y,z,w) \in \mathbb{R}^4 \mid x - y - 2z + w = 0\}$. The actual problem involves ...
2
votes
3answers
143 views

Two inner products being equal up to a scalar

I would appreciate a hint on the following problem: Let $V$ be a finite dimensional vector space over $F$. There are two scalar products such that: $$ \forall \ w,v \in V \ \Big(\langle ...
1
vote
0answers
57 views

What vector space is this?

Let $a,b,c$ be odd primes. In particular, $ab, ac, bc$ are all odd numbers. We can use this to our advantage, since then $\sqrt[ac]{x} : \Bbb{R} \to \Bbb{R}$ is well-defined and a bijection. Let ...
1
vote
3answers
38 views

If $A\ne V$ is a subspace of $V$ and $B$ is linearly independent subset of $V$, then $B$ can be completed to a basis of $V$

Let $A \ne V$ be a subspace of $V$ and $B$ a linearly independent subset of $V$. Prove that $B$ can be completed to a basis of $V$ with vectors from $V \setminus A$. OK, I started with: ...
0
votes
1answer
116 views

Proving orthogonal complement only contains 0 vector

Given the functions $v_n (x) = \frac{1}{\sqrt{2}}e^{i \pi n x}$, $n \in \mathbb{Z}$, I am to consider the orthogonal complement in $L^2 ([-a,a])$ of the vector space $V$ linearly generated by the ...
1
vote
2answers
90 views

Bilinear form question to prove isometry?

I am given $f$ to be a bilinear form on $V$, a finite dimensional vector space. It is symmetric with $f(v,w)=f(w,v)$ for all $v,w \in V$. Also $f(v,w) = 0 \forall w \in V$ implies that $v=0$. If I ...
5
votes
3answers
89 views

Generating a $n$-th dimensional vector orthogonal to $n-1$ linearly-independent vectors

Let us have $n-1$ linearly independent vectors $\vec{v}_{1},\dots,\vec{v}_{n-1}\in\mathbb{R}^{n}$, define the vector $\vec{w}$ as follows: ...
0
votes
2answers
284 views

Are elements of a basis pairwise orthogonal?

If $x_1,x_2,x_3$ are a basis for a real vector space $V$, then are $x_1,x_2,x_3$ pairwise orthogonal?
0
votes
3answers
57 views

Find a basis and the dimension of this vectorspace.

$U=\left \{ (x_1,...,x_n)\in\mathbb{R}^n:\sum^n_{k=1}x_k =0 \right \}$ My thoughts: (1) I think that $\text{dim(U)}=n-1$, but I'm not sure how to show that. (2) The basis needs to be a set of ...
1
vote
1answer
61 views

What are $S$ and $T$ in $C(T) = \{S \in L(V,V)\mid ST=TS\}$?

Let $C(T) \subseteq L(V, V)$ be the subset of all linear operators which commute with $T$, i.e.: $$C(T) = \{S \in L(V, V ) \mid ST = TS\}$$ Prove that: $C(T)$ is a subspace of $L(V, V)$, $C(T)$ ...
1
vote
3answers
258 views

vector problem in linear algebra

Prove that if $V$ and $W$ are three-dimensional subspaces of $\Bbb R^{5}$, then $V$ and $W$ must have a non-zero vector in common. So far I got $V = \{v_{1}, v_{2}, v_{3}\}$ and $W = \{ w_{1}, w_{2}, ...
1
vote
1answer
202 views

Equation of a plane containing 2 points

Find the equation of the plane containing the vectors $(1, 0, \sqrt{3})$ and $(1, \sqrt{3}, 0)$. The vectors are in standard position. I first get the vector between the 2 points $\vec{v}^{\ } = (1, ...
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votes
3answers
164 views

Linear algebra. Find a counter-example

this is the statement: if $\vec v_{1}, \vec v_{2} , \vec v_{3}, \vec v_{4}$ is a basis for the vector space $\Bbb R^{4} $, and W is a subspace of $\Bbb R^{4}$, then some subset of the $\vec v$ 's is a ...
3
votes
1answer
77 views

Find the basis and its dimention of a subspace

Given $S=\{x\in\Bbb R^4: x_1+x_2+x_3+x_4=0, x_2+x_4=0\}$ So (...) If $x_2+x_4=0 \implies x_2=-x_4$ then $x_1+x_2+x_3+x_4=0\implies x_1+x_2+x_3-x_2=0\implies x_1=-x_3\implies x_1=x_2=x_3=x_4=0$ ...
0
votes
1answer
175 views

$l_0$ is all sequences with finitely many non-zero terms. Show $W^\perp=\{y: <x,y>=0, x\in W\}=\{0\}$ where $W = \{x : <x,a>=0\}$.

Consider the inner product space $l_0$ consisting of all infinite sequences of complex numbers with only finitely many non-zero terms, with the inner product of $l^2$ (space of square summable ...
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votes
3answers
85 views

Span of vector in $\mathbb{R}^4$

Let $ a=(1,1,0,0), b=(0,1,1,0), c=(0,0,1,1), d=(1,0,0,1)\in\mathbb{R}^4$. Give a basis $\mathbf{B}$ for the intersection of the two linear spaces spanned by $a,b$ and $c,d$. Can anyone help me with ...
3
votes
2answers
108 views

$T$ be the operator from $C[0,1]$ to $C[0,1]$ defined by $Tf = f'+f''$. Show that the operator $T$ is unbounded.

$f \in C[0,1]$, the space of all continuous, complex-valued functions on $[0,1]$ with supremum norm. $\|f\|=\sup_{x\in[0,1]}|f(x)|$. Let $D$ be the set of $f \in C[0,1]$ such that the first ...
0
votes
1answer
49 views

product of vectors is zero then products of basis is.

Let $V$ be a vector space over a finite field $F_q$ and let $\{v_1, v_2, . . . , v_k\}$ be a basis of $V$. Show that the following two statements are equivalent: (i) $v\cdot v^\prime = 0$ for all $v, ...
0
votes
1answer
27 views

r(t) curve vector

Here's the problem: Integrate $f$ over the given curve. $$ f(x,y) = \frac{x+y^2}{\sqrt{1+x^2}}\qquad C: y=\frac{x^2}{2} \text{ from } (1,1/2) \text{ to } (0,0) $$ In the solutions manual it says ...
1
vote
0answers
77 views

Calculating Perpendicular Vectors from a single Look Vector

I have one vector that represents a 'look' direction. I need to calculate the 'right' direction and the 'up' direction. Say I have the vector( 0.79, -0.58, 0.188 ), I thought I could compute cross ...
2
votes
1answer
101 views

Getting rotation matrix from a vector

I have a vector pointing in some direction and I'm trying to find a matrix $M$ that rotates the vector $v_1=(1,0,0)$ to $v_2=(x,y,z)$, i.e., $M v_1 = v_2$. What is $M$ if $v_1$ and $v_2$ are known? ...
1
vote
0answers
64 views

Transformed Laplace “solution space”

From my own knowledge I can tell that when we take the Laplace transformation of a function we are in essence transforming our f(t) into a F(s). I've looked at several Q/A here asking for the ...
1
vote
3answers
244 views

what is the relationship between vector spaces and rings?

Can you show me an example to show how vector and scalar multiplication works with rings would be really helpful.
0
votes
1answer
44 views

Finding vector length based on parallell and orthogonal vectors

Do anyone know a simple way of finding the length of vector a in my figure? The known values are $(x_0, y_0), (x_1, y_1), (x_2, y_2), (x_3, y_3)$. (If you look closely, you can see that $f$-vector is ...
2
votes
2answers
22 views

For what other values of $a$ is $H_a$ a subset of $V$?

This is NOT homework, but review for a test. This is part b of a 2-part question, where part a was to show that $H$ was a subspace of $V$. I have done that part successfully and need help with the ...
2
votes
2answers
39 views

Determine a set H of all vectors (x, y, z) ϵ R³ that are L.C. of vectors U, V and W

Let me put the enunciation first: Let be the vectors $(U, V, W) \in \mathbb{R}^3: U = (1, 1, 5), V = (2, 1, 4), W = (-3,-1,-7)$, then determine the set $H$ of all vectors $(x, y, z) \in ...