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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48 views

Vector Spaces and Groups

I've just completed a course in linear algebra. I'm a physics undergraduate and I don't plan on taking an abstract algebra course. That said, I've been reading a little bit about it. As I understand ...
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2answers
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Norm of linear transformation: why restrict ourselves to $\|x\|\leq 1$?

If $f$ is linear transformation from a normed linear space $X$ into a normed linear space $Y$, and define its norm by $$\|f\|=\sup\{\|f(x)\|: x\in X, \ \|x\|\leq 1\}$$ My question is: why restrict ...
3
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2answers
42 views

Orthonormal basis . Can I have more than one basis for the subspace?

Required to find an orthonormal basis for the following subspace of R4 I know that to find the othonormal basis, it is required that i find the basis for the subspace, then I use Gram Schmidt ...
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1answer
23 views

Gram-Schmidt process in function subspace

I have a function space $\mathcal {F}([-1,1],\mathbb R)$ and the subspace $\mathcal{P_2}:=$ $(x\mapsto a_o+a_1x+a_2x^2| a_0,a_1,a_2 \in \mathbb R )$ for all polynomials with degree $\le2$. In this ...
5
votes
2answers
54 views

Determinant of exact sequence

Let $0 \to A \to B \to C \to 0$ be an exact sequence of vector spaces. I want to show that I have a canonical isomorphism $$\text{det}(B)= \text{det}(A) \otimes \text{det}(C).$$ Here, "det" refers ...
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2answers
54 views

checking if some vectors span $R^3$ that actualy span $R^3$

If we want to check if the following set of vector span $R^3$ (1,0,0) (0,1,-1) (0,4,-3) (0,2,0) then we forme an augmented matrix formed by the vectors which form the columns of the augmented matrix ...
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2answers
29 views

Dimension of a vector space below

I have to prove that the dimension of the vector space of real numbers over Q (rational numbers) is infinity. How can I prove? I have no idea.
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2answers
22 views

Co-ordinates of a vector in relation to the basis

Find the co-ordinates of the vector $u = (2,-1,4)$ of $\mathbb R^3$ in relation the basis $S = \{(1,1,1),(1,1,0),(1,0,0)\}$. Please could someone help/explain this to me, I'm doing revision for my ...
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2answers
53 views

“Non-linear” algebra

Linear algebra studies vector spaces and linear mappings between those spaces. What tools do we use for NON-linear mappings between vector spaces?
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33 views

Is a countable union of complete subspaces complete?

I would like to ask the following, which I wanted to use a part of my proof but couldn't determine if it's right: Assume $X$ is a normed space, and $(X_n)_{n\in \mathbb N}$ complete subspaces. Must ...
2
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2answers
23 views

Finding Bases from polynomials

Determine a basis from the following set of second degree polynomials. Does this basis span the space of the second degree polynomials? What is the dimension of the (sub)space that it spans? $$p_1 ( x ...
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2answers
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finding subspac of $R^4$ vectors described by equations

What is the dimension of the subspace of $R^4$ described with all vectors of the form $( w , x , y , z )$ that satisfy $$− 3 x + z = 0 , x + y + 4 z − w = 0$$ put me on the right track.
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0answers
39 views

Find the cardinality of the group $GL(2, \mathbb{Z/p^{n}Z}) $

Find the cardinality of the group $GL(2, \mathbb{Z/p^{n}Z}) $ for each prime $p$ and positive integer $n$ . What I know : Clearly if $n=1$ then cardinality of $GL(2, \mathbb{Z/pZ}) $is number of ...
0
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2answers
20 views

$4$ vectors can be a set of $\mathbb{R}^3$ subspace?

The question was $4$ vectors can be a basis for $\mathbb{R}^{3}$ ? I think from any $4$ vectors in $\mathbb{R}^3$ we can find a vector that is linear combination of others in $\mathbb{R}^3$ its ...
2
votes
2answers
31 views

Finding linear transformation such that $\operatorname{im} \phi = \ker \phi = \operatorname{span}(\alpha_1, \alpha_2)$

Here i am completely lost. I have to find a formula for linear transformation $\phi : \mathbb{R}^4 \rightarrow \mathbb{R}^4$ such that $\operatorname{im} \phi=\ker \phi = \operatorname{span} ...
0
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2answers
40 views

Show a $W$ is a subspace and find its dimension

Let $W=\{(a,b,c)\}\in \mathbb{R}^{3} : b=a+c \}$. Show that $W$ is a subspace of $\mathbb{R}^{3}$ and find $\dim(W)$. My solution is : Let $u=(a_1,b_1,c_1)$ and $v=(a_2,b_2,c_2)$ $\in W$ Then ...
0
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1answer
17 views

Show that the sequence $(A_n)_{n≥1}$ in $L(l_1)$ does not converge to zero

For any $n ≥ 1$, define a linear operator $A_n : l_1 → l_1$ by $$A_nx = (0, . . . , 0, x_{n+1}, x_{n+2}, . . .), ∀x = (x_1, x_2, . . .) ∈ l_1.$$ Show that For any $x ∈ l_1$, we have $\lim_{n→∞} A_nx ...
0
votes
1answer
97 views

Is the space of pairs with addition $(x,y) + (a,b) = (x+a+1,y+b+1)$ a vector space?

Prove the set $S =\{ (x,y) \mid x,y ∈ \mathbb{R} \}$ is a vector space with the operations of vector addition and scalar multiplication. $$(x,y) + (a,b) = (x+a+1,y+b+1)$$ $$\alpha(x,y) = (\alpha ...
1
vote
1answer
90 views

If the det of a set of vectors is zero, why does not span a vector space?

If we want to see if a set of vectors spans a vector space $V$, then lets say the set $A$ spans a vector space $V$ only If every linear combination of $A$ produces $V$, then $\text{Span}(A) = V$ ...
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votes
2answers
16 views

Relation between position vectors of a rectangle

I am given the position vectors of the rectangle a,b,c,d. I am supposed to prove that a.c=b.d (.=dot product) I tried representing the adjacent sides in terms of a,b,c,d since their dot product is ...
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2answers
25 views

The number of one-dimensional vector spaces in a field

Let $p$ be a prime number, $F=F_p$ a field with $p$ elements. $V$ is a vector space, $n$-dimensional over $F$. Calculate the number of one-dimensional vector spaces in $V$. I tried to solve it, but ...
0
votes
1answer
61 views

If $S\subset W$ and $W$ is subspace, is it ok to say $\operatorname{span}(S)\subseteq W$

I mean, if $S\subset W$ and $W$ is a subspace, then $S$ is either a basis for $W$ or at least spans some subset of $W$, therefore $\operatorname{span}(S)\subseteq W$. Is it ok? For finite sets it's ...
0
votes
1answer
32 views

Determinant on 3x3 matrix and above

When finding the determinent of a matrix, what is the rationale behind multiplying the entry along the row we are deleting from times the cofactor expansion? Also how does doing cofactor expansion ...
0
votes
2answers
29 views

Vector space theorem proof

Given $V$ a vector space, $\mathbf{u}$ is a vector in $V$ and $c$ is a real scalar then 1) $c\mathbf{0}=\mathbf{0}$ 2) $c\mathbf{u}=\mathbf{0}$ $\rightarrow$ $c=0$ or $\mathbf{u}=\mathbf{0}$ How to ...
0
votes
1answer
16 views

Find a basis for $U+W$ and $U\cap W$

Let $$W = \operatorname{span}([2,1,0,1], [0,0,1,0]) \\V = \operatorname{span}([1,2,1,3], [3,1,-1,4])$$ I need to find a basis and the dimension for $U+V$ and $U\cap V$. For $U+V$ I tried: $$U+V = ...
1
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1answer
10 views

Definition of onto for linear transformation

I had a question ask the following: "A linear transformation is onto if and only if the columns of its standard matrix form a generating set for its range." To me that seems true but the answer was ...
1
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1answer
37 views

If the null space contains only the zero vector, the map is one-to-one

How does finding out if the null space has only the zero vector prove one-to-one? One-to-one means that there are distinct images for each distinct vector input. $$\mathbb R^n \to \mathbb R^m$$ ...
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0answers
15 views

Searching for a definition for n-Dimensional rotation which is cosine-distance invariant

I am wondering if it is possible to define a rotation for an $n$-Dimensional space ($n=2,3,4,5,\dots$). Given any vector $\vec v$, and knowing that it should be rotated to ...
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2answers
42 views

proving a set V is a vector space (in one of the axioms)

If the set $V$ is defined by the points that go through the origin in $\mathbb{R}^2$ that satisfy the equation $ax+by=0$ then show $V$ is a vector space. Resolution:Proving that $V$ is closed under ...
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2answers
38 views

Having problem with Rotation and Reflection

Show the following, using matrices, combinations of linear transformations, and trigonometric identities. You must prove these in general – an example is not sufficient. (i) The combination of a ...
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0answers
29 views

Coordinate Basis of Transformation $T$

$B = ( v_1, v_2 )$ is the basis for $\mathbb{R}^2$ in terms of the vector and $v_1 = \begin{pmatrix} 1\\0 \end{pmatrix}$ and $v_2 = \begin{pmatrix} 0\\-1 \end{pmatrix}$ for a scalar $k$, the linear ...
1
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2answers
47 views

Show that $\{(1-a,1+a), (1+a,1-a)\}$ is linearly independent

In order to show that $$\{(1-a,1+a), (1+a,1-a)\}$$ is L.I. I did: $\beta_1(1-a,1+a)+\beta_2(1+a,1-a) = (0,0)\implies\\\begin{cases}\beta_1-\beta_1a + \beta_2 + \beta_2a = ...
2
votes
1answer
31 views

Proof that any set linearly independent has at most $n$ elements (when the vector space has basis with n elements)

My teacher gave us this proof today, but I don't know if I understood it entirely: Theorem: Suppose $V$ a vector space (finitely generated) over the reals. $$B = \{v_1,\cdots, v_n\}$$ where $B$ is ...
0
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1answer
30 views

Yet another n-Dimensional Rotation question: is there a definition for n-Dimensional rotation which is cosine-distance invariant? (TO CLOSE)

(Moved to Searching for a definition for n-Dimensional rotation which is cosine-distance invariant, flagged it to delete it) I'm wondering if there exists a rotation definition by which the vectors ...
0
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2answers
88 views

How do you take the inner product of a vector whose components have different units?

How do you take the inner product of a vector whose components have different units? For example, what is the inner product of $\langle1m, 1s\rangle$ and $\langle2m, 3s\rangle$?
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1answer
13 views

How to Find the Matrix of a Linear Map

If a map $f$ from the set of polynomials of degree 3 to the real numbers is given by $f(u) = u'(-2)$, how do I find the matrix that represents $f$ with respect to the bases $[1, t, t^2, t^3]$ and ...
0
votes
0answers
40 views

Shared Variance along non-orthogonal Axes

I have a dataset with $n$ points and $d$ dimensions in an $n \times d$ matrix $M$; I'm interested in the variance along various $d$-dimensional unit vectors $\{u_1, u_2 ... u_m\}$ which are not ...
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vote
2answers
36 views

Why two extension fields are isomorphic as vector spaces but not fields?

I understand that $\mathbb{Q}(\sqrt{2})$ and $\mathbb{Q}(\sqrt{3})$ are isomorphic as vector spaces but not as fields. However, I do not understand why that is true. What is happening when they are ...
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2answers
34 views

Question regarding dimensions of vector spaces

How can one prove the following ? Let $U,V$ be two vector spaces. $$\dim (U + V) + \dim (U \cap V) = \dim U + \dim V $$
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1answer
31 views

Verifying $(x_1,y_1) + (x_2,y_2) = (2x_1 - 2y_1,-x_1+y_1)$ is not vector space

I need to prove that this is not a subspace: $$(x_1,y_1) + (x_2,y_2) = (2x_1 - 2y_1,-x_1+y_1)$$ (and with the multiplication definied in a way that I will not write because the sum already fails) ...
2
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0answers
77 views

$\rm span(S_1) + \rm span(S_2) = \rm span(S_1 \cup S_2)$ for infinite sets

I have these two definitions of span: Span: Suppose a vector space $(V,+,\cdot)$, and $$S = \{u_1,\cdots,u_n\}$$ (and $S$ is a subset of $V$, not a subspace) ...
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1answer
21 views

Finding a basis for Kernel and Image of a linear transformation using Gaussing elimination.

This is a very fast method for computing the kernel/nullspace and image/column space of a matrix. I learned this from my linear algebra teacher but I haven't seen it mentioned online apart from this ...
0
votes
3answers
49 views

$\vec{r} \times (\vec{\omega}\times \vec{r})=r^2\vec{\omega}-(\vec{\omega}\cdot\vec{r})\vec{r} $

Show (in cartesian coordinates) that $\vec{r} \times (\vec{\omega}\times \vec{r})=r^2\vec{\omega}-(\vec{\omega}\cdot\vec{r})\vec{r} $ I am not really sure how to calculate this. Do I just assume ...
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vote
1answer
61 views

Problem on the dimension of a linear space

Suppose M(m,n) is the vector space which is composed of all $n \times m$ real matrices. Let $F(m,n)$, $n>m$, be the set of all $n \times m$ matrices and, for each element $u \in F(m,n)$, the rank ...
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1answer
34 views

Even degree polynomials form a vector space?? [closed]

Do the set of polynomials of even degree form a vector space?
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2answers
33 views

Is there a finite vector subspace over the reals?

I cannot think of a finite vector space over the reals, because we must sum these two elements and get a new element also in the vector space. And over the reals, I can't think of a sum that, at some ...
1
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1answer
35 views

Kernel and Image of linear transformation are supplementary

I read here that if $V$ is a vector space and $T : V \to V$ is a projection then $V = \text{ker}(T) \oplus \text{im}(T)$. I came up with a counterexample of the converse: $$S : \mathbb{K}_n[x] \to ...
0
votes
1answer
33 views

Prove that a function is a linear transformation.

Lets say that I have a vector space $A$ and a linear transformation defined as $f : A → A$. Now I have a function $g : A → A$ defined as $g(a) = bf(a)$ where $a\in A$ and $b \in \mathbb{R}$ is a ...
0
votes
2answers
48 views

Prove two commutative linear transformations on a vector space over an algebraically closed field can be simultaneously triangularized

Prove two commutative linear transformations on a finite-dimensional vector space $V$ over an algebraically closed field can be simultaneously triangularized. It is equivalent to show if $AB=BA$, ...
1
vote
1answer
21 views

Prove that $S(W)$ is Invariant subspace

Let $S, T: V\to V$ such that $ST=TS$. Let $W\subseteq V$. Prove that if $W$ is invariant subspace of $T$ then also $S(W)$ is invariant subspace of $T$. Let $w\in W$. $$T(S(w)) = S(T(w)) = S(w')$$ ...