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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3
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1answer
47 views

Does every isomorphism between $V$ and $V^*$ send some basis to its dual basis?

Suppose that I have a vector space isomorphism $\theta: V \to V^*$ where $V$ is any vector space (probably over $\mathbb{C}$ is required) and $V^*$ is its dual space. Is it always possible to find a ...
9
votes
3answers
164 views

Every invertible linear transformation can be perturbed a bit without destroying invertbility, Neumann series

Let $T: V \to V$ be any linear transformation on a real or complex vector space $V$. Show that there exists $\epsilon_0 > 0$ $($depending on $T$$)$ so that $I + \epsilon T$ is invertible for any ...
0
votes
0answers
14 views

Direct sum of two spaces

Let $\alpha_1=[1,1,0,1]$, $\alpha_2=[1,0,1,1], \alpha_3=[1,1,1,1],\alpha_4=[0,1,1,1]$ be a vectors from $\mathbb{R}^4$ let $U=span(\alpha_1, \alpha_2) \ and \ W=span(\alpha_3, \alpha_4)$ Check that ...
0
votes
0answers
8 views

Reference for work on abelian divisible groups $G$ such that for every $n \in \mathbb N , g \in G , \exists$ unique $x \in G$ such that $g=x^n$

Is there any work or reference in the literature about those abelian divisible groups $G$ such that for every $n \in \mathbb N , g \in G , \exists$ unique $x \in G$ such that $g=x^n$ ; I think then I ...
0
votes
2answers
18 views

Epimorphism of linear transformation

Find values of parameter t for which transformation is epimorphic: $\psi([x_1,x_2,x_3,x_4])=[x_1+x_2+x_3+2x_4,x_1+tx_2+x_3+3x_4,2x_1+x_2+tx_3+3x_4] $ When this transformation is epimorphic i.e. what ...
3
votes
1answer
64 views

Derivability of a function with an infinity of zeroes

Let $F$ be a normed vector space and $a\in F$. Is there a non zero function $f:\mathbb{R}\rightarrow F$, such that $f'(a)=0$ and $f$ is $0$ an infinity of times in any neighborhood of $a$ ? If not, ...
2
votes
1answer
30 views

Faulty proof that $V=U_1 \oplus W$ and $V=U_2 \oplus W$ implies $U_1 = U_2$

The question is as follows: Prove or give a counterexample: if $\ U_1, U_2, W$ are subspaces of $V$ such that $V=U_1 \oplus W$ and $\ V = U_2 \oplus W$, then $\ U_1 = U_2$. I happily ...
0
votes
0answers
11 views

Show that there exists a Hermitian form of signature $(p,q)$.

Let $K = \mathbb{Q}(\sqrt{-2})$ with $V_K = K^n$ considered as a $K$-vector space. Suppose $p, q \in \mathbb{Z}_{>0}$ such that $p + q = n$. Show that for any such $p$ and $q$ there is a Hermitian ...
1
vote
2answers
23 views

Prove vectors create a basis

Let $V$ be a vector space and $U,W,Z$ be it's subspaces where $V=Z \oplus U=Z\oplus W$. We know that $\beta_1,...,\beta_k$ is a basis of $U$ and $\beta_i=\gamma_i+\delta_i$ where $\gamma_i \in Z$ and ...
1
vote
0answers
22 views

Give the following linear transformation find values of parameter

Find values of parameter t for which transformation is epimorphic: $\psi([x_1,x_2,x_3,x_4])=x_1+x_2+x_3+2x_4,x_1+tx_2+x_3+3x_4,2x_1+x_2+tx_3+3x_4 $ When this transformation is epimorphic i.e. what ...
0
votes
0answers
36 views

Can anyone tell whether this vector space question is true or false? [duplicate]

If U and W are subspaces of a finite dimensional vector space V and V=U+W, then dimV≤dimU+dimW. we know that dimV=dim(U+W) and dim(U+W)> dimU+dimW, and therefore dimV>dimU+dimW. I think this is ...
1
vote
1answer
23 views

To find basis of subspace

Let V be subspace of $M_2 (R) $ consisting of all matrices with trace o and such that entries of first row add upto zero.To find basis for this My attempt i have posted here .according to me the ...
2
votes
1answer
46 views

Can anyone check these true and false statements about linear algebra?

For any square matrix $A$, the image of $A^7$ is contained in the image of $A$ I think this question is asking If $A^7x=b$, then $b$ must be in $A$ with some vector $y$ such that $Ay=b$. It Seems ...
1
vote
2answers
25 views

To find dimension of subspace

Let V be subspace of $M_n (R) $ be subspace ofall matrices such that entries in every row add upto zero and entries in every columm also add upto zero .Then i am to find its dimension . I have tried ...
0
votes
1answer
30 views

Orthogonality of remaining non-intersecting basis

Let $A$ and $B$ $\in \mathbb{C}^{4 \times 100}$ be matrices with null spaces $N(A)$ and $N(B)$ respectively. The dimensions of each null space is $96$ and I was able to find that they intersect in ...
2
votes
2answers
20 views

Orthographic projection in euclidean space

Let $E$ be a euclidean space with an inner product given by $$B =\left(\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 2 & -1 \\ 0 & -1 & 2 \end{array} \right) $$ in a basis ...
1
vote
3answers
30 views

Basis of a vector space is a maximal linearly-independent set?

If $V$ is a vector space of finite dimension over $F$, then a basis of $V$ is a maximal, linearly independent set in $V$. Is this conjecture true? If so, how to prove it?
0
votes
0answers
24 views

Understanding change-of-basis and linear operators

First of all , apologies in advance as this isn't so much as a question, but more check of my understanding. Suppose I have an $n$-dimensional vector space $V$ and a given basis $\mathfrak{B}= ...
0
votes
1answer
16 views

Product of $L_2$ norm of vectors

Is the $\sum \Vert b_k\Vert_2^2 \le\ge= \sum \Vert b_k\Vert_2^2 \Vert a_k\Vert_2^2$ ? where $b_k$ is a column vector and $a_k$ is a highly sparse row vector.
0
votes
1answer
37 views

What values must $\alpha$ be so that $F$ is an isomorphic linear transformation? (Bijective)

Let $F:P_2\to P_2$ where $P_2$ is a polynomial vector space with max grade of 2. $$[F]_B= \begin{pmatrix} \alpha & -1 & -1 \\ -6 & \alpha +1 & 0 \\ ...
0
votes
1answer
30 views

How to demonstrate a set is a real vector space (set governed by nonstandard operations)

I am really not that familiar with questions that ask you to work with a operation vector space, even less with the English terms for it. I am... quite lost. How would you prove that it is a real ...
1
vote
2answers
22 views

Proof that the kernel of an endomorphism to the power $n$ is a subset of the kernel of the endomorphism to the power $n+1$

I am expected to know how to prove the following but I can't seem to draw it out. Knowing that V is a Vector Space$$ T:V\to V $$ Prove the following $$ Ker(T^n)\subseteq Ker(T^{n+1}) $$ How ...
1
vote
1answer
15 views

Projection of vectors

Compute $:$ $proj_\vec y (\vec x)$ $\vec{x}_1=\begin{bmatrix} 2 \\ 3 \\ 4 \\ 5 \end{bmatrix}, \vec{y}_2 = \begin{bmatrix} 1 \\ 0 \\ -1 \\ 0 \end{bmatrix}$ Since the projection would be $:$ $(-2/0) * ...
1
vote
2answers
42 views

Will the value of $t$ affect the row, column, and solution spaces?

Consider A = $\begin{bmatrix}4 & 2\\t & 1\\3&t\end{bmatrix}$.Is the column space of $A$ the same for all t$?$Is the row space of $A$ the same for all $t$? Is the solution space of $Ax = ...
-3
votes
2answers
20 views

Problem on CR inequality on finite sum [on hold]

Let $f$ be a function from {1,2,3,....,10} to R, s. t. $(\sum_{i=1}^{10}|f(i)|/2^i)^2=(\sum_{i=1}^{10} |f(i)|^2)(\sum_{i=1}^{10}1/4^i)$ mark the correct statement. A. there are uncountably ...
-3
votes
1answer
58 views

Can the nullspace, the column space and the row space all be a line or a plane? [on hold]

Can the nullspace, the column space and the row space of a 4x3 matrix all be a line through the origin? Can the nullspace, the column space and the row space of a 2x4 matrix all be a plane through ...
1
vote
3answers
27 views

Showing that the magnitude of the difference of two vectors is larger than the difference of it's vector magnitudes

Long title. I have to prove (the problem itself suggests using Pythagorean theorem) the following inequality: $$\|u\|-\|v\| \le \|u-v\| $$ Vector magnitudes... How do you prove this in an ...
2
votes
2answers
31 views

The geometric meaning of a line plus a vector

Lets say we have $$ E = \{k(1,2,3)' + (2,9,-1)'\} \;\mathrm{with}\; k \in \mathbb{R} $$ we know that $k(1,2,3)$ spans a line in three dimensions, but what does the shape of $E$ look like. I think it ...
2
votes
3answers
105 views

What is $\Bbb{R}^n$?

I earlier asked this question The basis of a matrix representation. I now have a another question related to the same topic. The vector space $\Bbb{R}^n$ I have seen defined as all $n$-tuples of real ...
3
votes
2answers
63 views

The basis of a matrix representation

If I have the linear map $f:\Bbb{R}^n\rightarrow \Bbb{R}^m$ then we can write $f$ as like the following: $$f\left(\vec x\right)=A\vec x$$ Where $A$ is a matrix. I think $A$ is called the standard ...
1
vote
3answers
61 views

Question on the definition of vector spaces.

My question is perhaps useless, but I want to shed some clarity on this matter. I'm bothered by people that say a vector space is a "bunch of vectors". Or that a vector space "consists of ...
3
votes
1answer
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 ...
3
votes
2answers
30 views

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
votes
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 ...
0
votes
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
53 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 ...
1
vote
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 ...
1
vote
2answers
28 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.
1
vote
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 ...
1
vote
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?
1
vote
2answers
31 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 ...
1
vote
2answers
19 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 ...
0
votes
2answers
16 views

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.
1
vote
0answers
38 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
votes
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 ...
1
vote
0answers
17 views

Questions regarding Banach spaces [closed]

Problem 1: Let $X$ be a Banach space and $I ∈ L(X)$ be the identity operator. Determine the action of the operator $e^I$ on $X$. Problem 2: Let $X$ be a Banach space and $A ∈ L(X)$ be a bounded ...
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
votes
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
votes
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 ...