Tagged Questions

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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1answer
13 views

Minimal polynomial in $T$-invariant subspace

I am stuck on the following problem. Problem: Let $V$ be a finite dimensional vector space over field $F$ and $T$ a linear transformation from $V$ to $V$. $W$ is an invariant subspace. Let $h_1$ be ...
1
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3answers
26 views

Help understanding a proof about vector spaces

The exercise goes like this: -Let $W= {(x,y,z)|2x+3y-z=0}$ Then $W\subseteq\mathbb{R}^3$, find the dimension of $W$. -Find the dimension $[\mathbb{R}^3|W]$ This was a problem from my algebra exam, ...
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0answers
15 views

Proof Explanation: Vector Space of Polynomials with Average Value 0 around a circle

The question is from Putnam 2009 B4. Problem: Say that a polynomial with real coefficients in two variable, $x,y$, is balanced if the average value of the polynomial on each circle centered at the ...
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3answers
18 views

Determining if a set is in the subspace of a continuous function

Let $A={\rm span}\{\cos^2x,\sin^2x\}$ be a subspace of the set of functions $C[0,\pi]$, for each of the following functions in $C[0,\pi]$, determine whether or not it is in $A$. $f(x)=1$ ...
1
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2answers
52 views

Prove $W \cap W^\perp =\{\vec{0}\}$

If $W$ is a subspace of $\mathbb{R}^n$, then $W^\perp = \overline{W} = \{v \cdot w = 0, \forall w \in W\}$ Prove $W \cap W^\perp = \{\vec{0}\}$. How do I fully prove this intersection is ...
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1answer
18 views

Need help regarding Subspace of matrix and its basis

I need some kind of hint to get me going for this question as I'm so lost at it. Any sort of help would be appreciated. Let E be the set of all 2x2 matrices that have $v={(1,-1)}$ as an eigenvector. ...
1
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1answer
8 views

Find all unit vectors in the plane determined by vectors u and v that are perpendicular to the vector w.

Find all unit vectors in the plane determined by vectors u=(0,1,1) and v=(2,-1,3) that are perpendicular to the vector w=(5,7,-4). This is the question. I found the plane that determined by u and v, ...
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0answers
16 views

Check if set of functions is a basis of space

Let $f_a \in R^R$ be function given by $f_a(x)=1$ if $x=a$ and $f_a(x)=0 $ if $x \neq a$ for $a \in R$ Decide if set of functions $f_a$ is a basis of space of functions $R^R$ ? I think I know how to ...
2
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1answer
18 views

Do two isomorphic finite field extensions have the same dimension?

If $E = F(u_1, \cdots u_n) \cong \bar{E} = F(v_1, \cdots v_m)$ then do the two extensions necessarily have the same dimension over $F$?
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0answers
23 views

Can we show it without involving that $V=V^{**}$ are canonically isomorph?

My text proves the following Theorem. Let $V$ be a vector space over $F$ and $B=\{ v_1, \ldots , v_n \}$ a basis of $V$. Then there is exactly one basis $B^*=\{ f_1, \ldots , f_n \}$ of $V^*$ with ...
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1answer
44 views

Verifying the axioms of a vector space for $V =\{(a,b):a,b\in\mathbb R \}$ with unusual scalar multiplication [on hold]

Let $V =\{(a,b):a,b\in\mathbb R \}$. Addition in $V$ is $(a_1,b_1) +(a_2,b_2) = (a_1+a_2, b_1+b_2)$ and scalar multiplication is $k(a,b) = (ka, 0)$. Is $V$ a vector space? Why? I'm mostly lost ...
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0answers
15 views

1st isomorphism theorem for linear transformations (algebra)

For a field K, U' and U'' are vector subspaces of a vector space U over K. It needs to be proven that the transformation φ: U' →(U' +U'')/U'', u' 􏰀→u' +U'', is a surjective linear transformation, ...
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0answers
43 views

Linear algebra and geometric insight: a rigorous approach to vector spaces, matrices, and linear applications

Could you point out some references (undergraduate level) that give a geometric understanding of vector spaces, matrices, and linear applications? As far as I know, many textbooks start with ...
2
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1answer
32 views

Which Field Operators Construct the Vector Space

Question 14 in F-I-S section 1.2 asks: Let $\mathbf{V}=\{(a_1,a_2,\ldots ,a_n)\colon a_i\in \mathbb{C}$ for $i=1,2,\ldots n\}$; so $\mathbf{V}$ is a vector space over $\mathbb{C}$. Is $\mathbf{V}$ ...
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1answer
15 views

“Absolutely equal” linear functionals and collinearity

Let $(X,\|\cdot\|)$ be a normed vector space over $\mathbb C$ and let $X^*$ denote its dual (i.e., the space of all continuous linear complex-valued functions over $X$). Suppose that $f,g\in X^*$ ...
1
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1answer
33 views

Prove that the subset $X$ of a normed vector space $(V,\|\cdot\|)$ is complete.

My subset $X$ has the Bolzano-Weierstrass property and I need to prove that $X$ is complete in the sense that every Cauchy sequence in $X$ converges to a point in $X$. I know that having the ...
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0answers
30 views

Independence in Banach space

Everyone knows one of the basic theorems in linear algebra: $k+1$ vectors can't be linear independent in the span of $k$ vectors. Also, it's pretty easy to prove that there is no uncountable system of ...
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0answers
7 views

Axes of rotation, recursive tree branching and GLrotate (computer graphics)

The question is to solve a computer graphics problem, but is essentially a vector math problem so I think it belongs here. My problem is this: a recursive tree is being generated for n iterations ...
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0answers
9 views

Why is the following progression allowed?

I'm looking at a proof of a thing related to vector spaces and in that proof we have the following progression: x$^{H}$($\alpha$y + $\beta$z) = $\alpha$x$^{H}$y + $\beta$x$^{H}$z where x, y, z are ...
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5answers
51 views

linear independence with $\sin x, \cos x$

I don't know why $\sin x$ and $\cos x$ are lineary independent since if we take linear combination $a\cdot \sin x + b \cdot \cos x=0$ and for $a=\sqrt{3}$ and $b=1$ and $\displaystyle x=\frac{\pi}{6}$ ...
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1answer
16 views

Finding vector when conditions are given

Given subspace (of $\mathbb{R}^4$) $V= \rm span ([2,3,1,2], [3,2,2,3], [1,-1,1,1]) $ For $\beta_1=[1,1,1,1], \beta _2=[2,-1,1,2]$ desribe set of all vectors $[b_1, b_2] \in \mathbb{R}^2 $ such that ...
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2answers
69 views

Is the statement “the empty set is a subspace of every vector space” true of false?

Is this statement true or false, and why? The empty set is a subspace of every vector space.
1
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1answer
30 views

Find $t$ such that (subspace)

For $t\in\mathbb{R}$ a subspace of $\mathbb{R}^4$ is given $V_t={\rm span}([t, -1, 2,-1],[1,-1,-1,1])$. Find all $t\in\mathbb{R}$ such that $V_t={\rm span}([4, -3, 0,1],[1,0, 3,-2])$. EDIT: Still ...
0
votes
1answer
15 views

Finding base vectors

How to find base vectors of such a subspace given by the following equation: $W=\{[x_1,x_2,x_3,x_4] \in\mathbb{R}^4 : 2x_1+x_2-x_3-x_4=0 \}$
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0answers
32 views

Suppose $V$ is subspace of a Hilbert Space $\mathcal H$. Show the identity $\bar V = (V^{\bot})^{\bot}$

Suppose $V$ is subspace of a Hilbert Space $\mathcal H$. Show the identity $\bar V = (V^{\bot})^{\bot}$. I've already proved that if $U$ is a closed subspace then $U = (U^{\bot})^{\bot}$. I also ...
0
votes
1answer
15 views

Let $S$ be a vector space. I want to show that for every subspace $U$ of $S$, the closure $\bar U$ is again a subspace of $S$.

Suppose $S$ is a vector space with norm $|| \ ||$ and $\rho$ the corresponding metric. I want to show that for every subspace $U$ of $S$, the closure $\bar U$ is again a subspace of $S$. I've proven ...
1
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2answers
19 views

What does dimension of polynomial mean?

So, I know that the vector space of polynomials with degree $n$ has dimension $n+1$. What does this exactly mean? I'm asking specifically because of the following question (from Putnam and Beyond): ...
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1answer
32 views

To prove $V$ Is not a vector space and my attempt

Let V be set of all pairs (x,y) of real numbers , And F b field of real numbers Define $c(x,y)$ = $( |c|x , |c|y)$ Addition defined as usual but it is not main concern here What i have done is ...
0
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1answer
23 views

How is this expression well-defined?

I am going through the book "Introduction to Tensor Product of Banach Spaces" by Raymond Ryan. The tensor product of vector spaces is introduced in the first chapter which I briefly outline now. Let ...
1
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1answer
14 views

Finding vector from a given subspace

Given the subspace described by those three vectors: $W=lin([1, 0, 2, 4], [0, 1, 2, 3], [0, 1, 0, 1] ) $ Now i want to pick up any vector that is located in this subspace. How should i do this?
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0answers
29 views

Is this a vector space

Can you prove that this is a vector space?
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0answers
19 views

A problem related to Normal Operator and Self- adjoint [on hold]

Let T be a normal operator on a finite dimensional linear inner product space V whose characteristic polynomial splits. Prove that V has a orthonormal basis of eigen vectors of T. hence prove that T ...
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1answer
27 views

A problem concering basis of a vector space [on hold]

Let $V$ be a vector space having dimension $n$, and let $S$ be a subset of $V$ that generates $V$. (a) Prove that there is a subset of $S$ that is a basis for $V$. ($S$ need not be finite.) (b) ...
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1answer
32 views

Coordinates of vectors in bases

Two vectors from the standard basis are a = (1,0,1) and b = (1,1,1). What are the coordinates of these vectors in the basis {(1,2,3),(2,3,1),(3,0,1)}. I am not even sure how to answer this question. ...
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0answers
11 views

Product of self adjoint transformations

If $A$ and $B$ are linear transformations such that $A$ and $AB$ are self-adjoint and such that $\ker (A) \subset \ker (B)$, then does there always exist a self-adjoint transformation $C$ such that ...
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0answers
46 views

Dimension of image of a skew symmetric map is even

If $A$ is a skew-symmetric linear transformation on a finite-dimensional Euclidean space, then rank $\rho(A)$ of $A$ i.e., the dimension of image of $A$ is even. I am trying for a geometric proof of ...
1
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1answer
20 views

Eigenvalues of linear operator TS and ST for infinite dimensional space

Here is the original problem: Let $S$ and $T$ be linear operators on a finite-dimensional vector space $V$. Show that $TS$ and $ST$ have the same eigenvalues. I can prove it. However, my question is: ...
1
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1answer
23 views

show that if S(finite) spans V(finite), (but not basis) then S can be reduced to a basis for V by removing appropriate vectors from S

problem 18: Suppose that S be a nonempty finite set of vectors in a finite dimensional Space V. Use problem 16 to show that if S spans V, but is not a basis for V, then S can be reduced to a basis for ...
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1answer
22 views

Basis for solution space?

For the matrix: $$ \begin{bmatrix} 1 & 0 & 2 & | & 0 \\ 0 & 1 & 3 & | & 0 \\ 0 & 0 & 0 & | & 0 \end{bmatrix} $$ ...
1
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2answers
38 views

Exponential objects in $k$-$\mathbf{FDVect}$

In my differential geometry class we've now moved onto algebraic/differential forms and to begin the section we're doing a quick and easy review of dual vector spaces. On a problem sheet I am ...
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3answers
48 views

Cardinality of the basis for $\mathbb{R}$ over $\mathbb{Q}$? [duplicate]

This question came up as a discussion I had with a friend. Clearly, the basis is not of finite cardinality since that would imply the set $\mathbb{R}$ has the cardinality $\aleph_{0}$ which is false. ...
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0answers
14 views

Set of all linear combinations of the vectors is a subspace

Let $T =$ {$v_1, v_2, ... , v_n$} be a set of vectors in $R^m$, and show that the set of all linear combinations of the vectors in $T$ is a subspace of $R^m$. And that much I able to find, $T =$ ...
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votes
1answer
30 views

How to prove the law of sines of elementary geometry using the vector product?

Let $V$ be a $3$-dimensional Euclidean vector space with scalar product $(.|.)$. Use the vector product to prove the law of sines of elementary geometry, which says: If $α$, $β$, $γ$ are the angles ...
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3answers
38 views

How to check that this is an orthogonal linear map with $\det (A) = 1$, so it is a rotation?

$V$ is a $3$-dimensional Euclidean vector space with scalar product. Let $(e_1,e_2,e_3)$ be an ordered orthonormal basis of $V$ and let $A$ be the permutation operator defined by $$A(e_1) = e_2, ...
0
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1answer
5 views

Calculate 1-norm of a vector using another matrix or vector

Let's say I have a vector a. I would like to construct a matrix or vector b such that if I multiply ...
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0answers
12 views

two problems with finding equations of vector subspace and vector space

1) If vectors: $(1,-1,1,-1,1),(1,1,0,0,3),(3,1,1,-1,7),(0,2,-1,1,2)$ are in $R^5$ then describe the least vector subspace by equations. Here I have to solve the matrix and here is an answer ?? 2) ...
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0answers
21 views

How to find the axis of the rotation?

$V$ is a $3$-dimensional Euclidean vector space with scalar product. Let $(e_1,e_2,e_3)$ be an ordered orthonormal basis of $V$ and let $A$ be the permutation operator defined by $$A(e_1) = e_2, ...
1
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3answers
25 views

Similarity of linear transformations

Suppose there are two linear transformations $A$ and $B$ on the same finite dimensional vector space $V$, such that $\dim Im(A) = \dim Im(B)$. Is it always true that they are similar. What about the ...
2
votes
1answer
44 views

Vector space and algebraic closure of a field

I hope you can help me with these questions, I can't really come up with a solution! Let $V_k$ be a vector space of dimension $n$ over a field $k$. Let $K=\bar k$ be the algebraic closure of $k$. A ...
2
votes
1answer
44 views

The norm of operator $\mathscr{L}$ on the finite-dimensional vector space $V$ equals the norm of operator restricted by some Invariant subspace.

The norm of linear transformation $\mathscr{L}$ on the finite-dimensional vector space $V$ over $\mathbb{R}$ with standard inner product equals the norm of linear transformation $\mathscr{L}$ ...