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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Angle between two vectors on manifold

I'm parallel transporting a vector along a curve and trying to calculate how much this vector rotates relative to the curve's tangent vector. So if the path is a geodesic then I will get an answer of ...
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2answers
111 views

Relationship between the singular value decomposition (SVD) and the principal component analysis (PCA). A radical result(?)

I was wondering if I could get a mathematical description of the relationship between the singular value decomposition (SVD) and the principal component analysis (PCA). To be more specific I have ...
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1answer
28 views

Elementary problem about Tensor product and Kronecker product defined by linear map

I have some perplexities when I reading references about tensor product. My main question is: How to define the tensor product between two vectors? It is clearly to define the tensor product ...
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1answer
31 views

Proof that Vector Space in Domain of Linear Map is a Direct Sum

I'm working through problems in Linear Algebra just for fun and I am getting stuck on Axler 3.4. Suppose that $T$ is a linear map from $V$ to $\mathbf{F}$. Prove that if $u \in V$ is not in $null\ ...
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1answer
46 views

How to interperet calculus thing

I have $\nabla \times (f\mathbb{F})$ where $f$ is a twice continuously differentiable scalar field and $\mathbb{F}$ is a twice continuously differentiable vector field. Is it right to interpret $f$ ...
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41 views

Vector Space Verification

I just took an exam asking me if the following are a vector space over $\mathbb{R}$ assuming that the set of all real valued functions on the interval $[0,1]$ is a vector space with theoperations ...
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1answer
17 views

Show that $\mathcal{C}(A)$ is the smallest convex of $E$ containing $A$.

Let $E$ be a $\mathbb{R}$-vector space, and $A$ a nonempty subset of $E$. Show that $$\mathcal{C}(A) = \biggl\{\sum \limits_{k=1}^n \lambda_kx_k \biggm| n \in \mathbb{N}^*, (x_1,\dots,x_n) \in ...
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3answers
57 views

Let $V $be a vector space. Prove/Disprove: There is a norm $\|\cdot\|$, such that all subsets of $V$ are open sets in $(V,\|\cdot\|)$.

The Assignment: Let $V$ be a vector space over $\mathbb{R}$ with $V \not= \{0\}$. Prove or disprove: There is a norm $\|\cdot\|_d$ on $V$, such that all subsets of $V$ are open sets in ...
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1answer
31 views

Why do special solutions of $Ax=0$ form a basis for null-space of $A$?

I read somewhere that The $n-r$ special solutions of a $m \times n$ matrix with rank $r$ form a basis for its null-space. If we consider the general RREF for the given matrix, it has the form: ...
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1answer
29 views

linear space proof

The question is that V is the span of these vectors in the diagram b2,b3,b4. Please help me in this problem, I know all the theory that for it to be a linear space it should be closed under addition ...
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3answers
34 views

Definition of linear independence when $v_i=0$

I have read that linear independence occurs when: $$\sum_{i=1}^n a_i v_i =0$$ Has only $a_i=0$ as a solution, but what if all $v_i$ were $0$ then $a_i$ could vary and still yield $0$. Does that mean ...
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1answer
23 views

finding the Rank and basis of null space of this matrix

Please help me with this question. The question is to find the rank of the matrix and then the basis of the null space, I first put the matrix A in reduced row echelon form and then I wrote the ...
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1answer
84 views

How to prove $ |\langle u,v\rangle| \leq ||u||||v||$

How to prove $ |\langle u,v\rangle | \leq ||u||||v||$ Note: I have given this many attempts so don't downvote due to lack of effort, refer to edit history for evidence of said effort
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3answers
115 views

Subspace of a finite dimensional inner product space, independence of basis choice

Let $W$ denote a subspace of a finite dimensional inner product space $V$, and let $$\beta = \{w_1,w_2,\dots,w_r\}$$ denote an orthogonal basis for $W$. For any $v\in V$ define $$proj_{\beta}v = ...
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0answers
26 views

Proving that combinations of these linear spaces are not linear spaces [duplicate]

The vectors $\mathbf{b}_1$, $\mathbf{b}_2$, $\mathbf{b}_3$, $\mathbf{b}_4$ are defined as follows: $$ \mathbf{b}_1 = \begin{pmatrix}1\\0\\0\\0\end{pmatrix}, \mathbf{b}_2 = ...
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2answers
68 views

Linear Space in Vector Spaces question.

How do I do the first part of the question where they say V1 U V2 is not a linear space, please help my exam is very close. In the marking scheme it says it's not closed under addition but can ...
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0answers
15 views

Sufficient conditions for RTree

What is the sufficient screening criteria of a space for the possibility to use R-Tree spatial index on it? I cannot apply it to a space with just Jaccard distance as the metric. As I suppose the ...
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1answer
22 views

Is there any example of usage for a vector space over the field of formal Laurent series?

The formal Laurent series over a field is a field. Is there any example where vector spaces over that field occur naturally?
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1answer
12 views

Calculating a basis in $\mathbb{R}^4$.

Have the subspace of $\mathbb{R}^4$ $$W = \{(x,y,z,w) \in\mathbb{R}^4 : y - w + z = 0\}$$ Calculate a basis for $W$, and then find an orthonormal base from that. The basis, from the ...
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1answer
23 views

How does full row rank imply column space is $R^m$ for a $m \times n$ matrix?

From Gilbert Strang's textbook Introduction to Linear Algebra (p.159) Every matrix with full row rank has these properties $Ax=b$ has a solution for every right side $b$. The column ...
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1answer
32 views

Calculating a basis given some constraints.

Have a vector space formed by the vectors $(x_1,x_2,x_3,x_4)$ that satisfy $$\begin{cases} x_1+x_2-x_3-3x_4=0\\ 2x_1+x_3-2x_4=0 \end{cases}$$ Find a basis and also the dimension of $S$. ...
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1answer
28 views

Does linear dependency have anything to do when determining a span?

Q: Does $\{(1,1) , (2,2)\}$ span $\mathbb{R}^2$? A: No, because they are linearly dependent. I agree that it doesn't span $\mathbb{R}^2$, but from my understanding, linear dependency has ...
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3answers
23 views

Proving that a subset is a subspace by showing a scalar combination.

Prove that: $$S = \left\{\left(\begin{matrix}a & b \\ c & a\end{matrix}\right) \ / \ a,b,c \in \mathbb{R}\right\} \subset M(2,\mathbb{R})$$ Answer: $S$ is a scalar ...
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1answer
40 views

When proving if a subset is a subspace, can I prove closure under addition and multiplication in a single proof?

I'm learning about proving whether a subset of a vector space is a subspace. It is my understanding that to be a subspace this subset must: Have the $0$ vector. Be closed under addition (add two ...
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1answer
32 views

Using rgb triplets in “scalar” math

So I'm reading a computer graphics research paper and I'm confused as to how to interpret certain formulas that are being used. In the paper, a value σ is defined as an RGB triplet. Later on, that ...
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2answers
38 views

Adding squared numbers to find magnitude

I'm using Pythagoras' Theorem to try and work out the magnitude of a vector: vector = (-3, 7) magnitude = $\sqrt{ -3^2 +7^2 }$ magnitude = $\sqrt{ -9 + 49 }$ Now here's the thing, I ...
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2answers
36 views

Function for diagonalizing a vector.

I was playing around whith the idea of what operation (function) should I perform (apply) over a vector $\mathbf{a} = (a_1,a_2, \ldots, a_N)^T \in \mathbb{R}^N$ to come up with the following matrix: ...
2
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0answers
57 views

kernel space of linear combination of matrices

Suppose $A$ and $B$ are $N\times N$ matrices so that for every $x$ and $y$, $xA+yB$ has a kernel of dimension at least $2$. Is it necessarily true that $\ker(A)\cap\ker(B)$ has dimension at least ...
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1answer
18 views

linear combination vectors into one vector

write $x(a_1,a_2,a_3)+y(b_1,b_2,b_3)+z(c_1,c_2,c_3)$ as $Y(a_1,a_2,a_3,$ $b_1,b_2,b_3, $ $c_1,c_2,c_3)$ ^as a 3x3 matrix for a suitable Y?
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1answer
25 views

Dimension of vector space (count 0 or not)

Do you count 0 in the dimension of a vector space? Eg. If $V_\lambda$ is the eigenspace of a certain function $f$, which has eigenvectors corresponding to $\lambda$ of $v_1, v_2, v_3$ then the basis ...
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1answer
49 views

Is there a specific method to finding a basis for vector spaces over $\mathbb{Q}$ ?

I am stuck on the first one but there are 5 questions on this so I really need help with the process. If anyone can help with any of the following. i) Find a Basis for the field K = ...
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2answers
73 views

How to find a basis for a tricky 2x 2 matrices vector space

Consider the vector space of 2 x 2 matrices :\begin{bmatrix}a&b\\0&c\end{bmatrix} such that a and c are rational numbers and b is a real number with rational numbers as the field of this ...
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1answer
15 views

Understanding the basis term

Consider: $$\left( {\matrix{ 0 & 1 & 2 \cr 0 & 0 & 0 \cr } } \right)$$ I want to find a basis for the row-space of the matrix above. One might say $$B = \left\{ {\left( ...
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0answers
16 views

Show that there exists f-stable subspace for certain conditions

Let K be field. Assume that the characteristic of $K$ is not equal to $2$. Let $f:V \rightarrow V$ be a linear operator such that $f(f(v))=v$ for any $v \in V$. Show that there exist f-stable ...
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2answers
37 views

Prove that the linear space of polynomials with root $\alpha \in \mathbb{R}$ is a subspace of $\mathbb{R}[x]_n$

Prove that linear space of polynomials having root $\alpha \in \mathbb{R}$ is a subspace of $\mathbb{R}[x]_n$. It's also required to find basis and dim of that subspace. I recently started learning ...
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3answers
51 views

Find eigenvector of the linear operator

Task is to find an eigenvector of the following linear operator: $f \to \int^{x}_{-x} f(t)dt$ in the linear span $\langle cos(x), sin(x), ...,cos(mx),sin(mx)\rangle$. I know how to find eigenvectors ...
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1answer
26 views

Lattice inside a finite dimensional vector space

I have an integral domain $R$ and its field of fractions $K$. Let $V$ be a finite dimensional $K$ vector space. Let $M$ be a finitely generated $R$-module contained in $V$. Why is $K\cdot M=V$ ...
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2answers
46 views

The equation of the plane in five-dimensional space

Given a five-dimensional space. There are three points (coordinates) and need to find the equation of a plane through 3 points. How to do this? $$B(1,1,0,1,1)$$ $$C(8,7,3,1,4)$$ $$D(1,0,-1,3,-3)$$
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3answers
90 views

Can such a set of vectors exist?

Is it possible, given $n\geq 3,$ to find $\{\mathbf{x}_1,\;\mathbf{x}_2,\;\dots,\;\mathbf{x}_n\}\subset\mathbb{R}^n$ satisfying: $\mathbf{x}_i,\mathbf{x}_j$ linearly independent for any distinct ...
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2answers
22 views

Scalars determine a vector in inner product space.

Let $V$ be a finite dimensional inner product space over $k$ with basis $\{v_1,\dots,v_n\}$ and inner product $\langle \cdot,\cdot\rangle$. For any $\alpha_1,\dots,\alpha_n\in k$, there exists a ...
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1answer
28 views

“$q$-linear envelopes” of $\mathbb{F}_p$-subspaces

Let $V$ be a vector space over an algebraically closed field $k$ of characteristic $p>0$, and denote by $V_q$ the vector space obtained from $V$ by restricting scalars to $\mathbb{F}_q$, where ...
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1answer
30 views

Proof Green's theorem $F(x,y)=(x-y)i+xj$

I was reading on Green's theorem and have appreciated the concept. Given a question, I think, I can solve it.But I came across a question that reads: Verify the Green's theorem for the vector given ...
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48 views

What is the operation inverse to vectorization (vec operator)?

There is a well knows vectorization operation in matrix analysis $\mbox{vec}$: https://en.wikipedia.org/wiki/Vectorization_%28mathematics%29 I've vectorized my matrix equations, did some ...
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29 views

Tensor Product over a field

This question appeared in my exam and I could not solve it. Let $L$ be a field, $K$ subfield of $L$. Assume that dim$_K(L)$=$m$, and $V$ be a $L-$vector space amd dim$_L(V)=n$. If as usual ...
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0answers
19 views

Separating convex sets in a tvs $X$.

I got doubt with the proof of this theorem. Let $X$ be a tvs, $A,B \subset X$ with $A$ an open convex set and $B$ convex such that $A \cap B = \emptyset$. Then there exists $f \in X^*$ (where ...
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0answers
39 views

How do i show that something has the structure of a vector space and how do i prove something to be linearly independent?

Im not sure what to do.....where do i begin? $\quad$ Throughout, $V$ denotes a vector space of finite dimension $n$ over the field $F$, and $\tau$ an element of ${\cal L}(V).$ Recall the structure of ...
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1answer
12 views

Lines perpendicular to vectors- are they similar triangles?

Please excuse my horrible vector drawing skills. Let us first assume that we have a third vector, called $\Delta V = V_2 - V_1$ Now, these three vectors make a triangle, $V_1, V_2, \Delta V$. Let ...
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1answer
48 views

How do I rewrite vectors in other basis' given change of coordinate matrices?

$\displaystyle β= \begin{bmatrix}2\\2\\\end{bmatrix}$,$\displaystyle \begin{bmatrix}4\\-1\\\end{bmatrix}$ $\displaystyle C= \begin{bmatrix}1\\3\\\end{bmatrix}$,$\displaystyle ...
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2answers
80 views

Find a matrix such that $Ax=0$

Let $$W = span\left\{ {\left( {\matrix{ 1 \cr 0 \cr 0 \cr 1 \cr } } \right),\left( {\matrix{ 0 \cr 2 \cr 1 \cr { - 1} \cr } } \right)} \right\}$$ I was asked ...
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1answer
31 views

Finding a basis for intersection of two subspaces

Let two subspaces of $V=\mathbb{R}^4$: $$w1 = \left\{ {\left( {\matrix{ 1 \cr 1 \cr 1 \cr 1 \cr } } \right),\left( {\matrix{ 1 \cr 0 \cr 2 \cr 0 \cr } } ...