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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Orthocenter of triangle given by vectors in $3$-dimensional space [on hold]

Given: $B=(1,0,0), D=(0,1,0), E=(0,0,1)$. Question: find the coordinates of the orthocenter $L$ around triangle $BDE$. With the formula used
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1answer
13 views

Determining Line Integrals from a Graph and Vector Field (Image Included)

Consider the vector field: $$F=\left(\frac{2xy-2xy^2}{\left(1+x^2\right)^2}+\frac{8}{13}\right)i+\left(\frac{2y-1}{1+x^2}+2y\right)j$$ Determine $$\int_C F\cdot dr$$ where $C$ is the path ...
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vector-valued function of the given curve from two given points

So I have to fin a vector-valued function of the portion of the parabola $z = 4y^2$ on the yz-plane from the point $(0, −1, 4)$ to $(0, 2, 16)$ I don't even know where to start from this, if i get ...
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29 views

Show that there is a vector $w$ in ${\rm ker}\ (T)$ such that $v=u+w$

Suppose $U$ and $V$ are vector spaces such that $T:U\rightarrow V$ is a linear map. Suppose also that $u$ and $v$ are vectors in $V$ such that $f(u)=f(v)$. Show that there is a vector $w \in \rm ...
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24 views

Basis of orthogonal complement subspace [duplicate]

Let $A$ be the matrix $$ \begin{pmatrix} 1 & 1 & -1&-1 \\ 1 & 2 & -2 & 1 \\ \end{pmatrix} ,$$ let $W$ = ker $A$ and let $W^\bot$ be the ...
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1answer
15 views

Proofs involving orthonormal basis

Suppose that $V$ is an inner product space. (a) Show that if $\{e_1, . . . , e_n\}$ is an orthonormal basis for $V$ , then $$||v||^2=\sum_{i=1}^{n}|\langle v|e_i\rangle|^2\quad \quad \text{for every ...
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1answer
35 views

Spans of Orthogonal complements

Let $A$ be the matrix $$ \begin{pmatrix} 1 & 1 & -1&-1 \\ 1 & 2 & -2 & 1 \\ \end{pmatrix} ,$$ let $W$ = ker $A$ and let $W^\bot$ be the ...
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1answer
20 views

Is the set $V$ = { $([t], [g], [t], [j]): t,g,j∈$Z$,[2t+j] = [0]$} a subspace of vector space $(\mathbb Z_3)^4$?

Is the set $V$ = { $([t], [g], [t], [j]): t,g,j∈Z,[2t+j] = [0]$} a subspace of vector space $(\mathbb Z_3)^4$? I am inclined to think that it is a subspace. However, I cannot find any basis for the ...
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Rotation Matrix in domain and co-domain basis

I was asked t o derive the rotation matrix counterclockwise with given angle in different domain and co-domain basis. Using what we know from trigonometry I derived the Rotation matrix as: R(Q) = ...
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20 views

Skew-symmetric non-degenerate bilinear form

If we do symplectic linear algebra on a finite-dimensional vector space $V$, then what does $$\omega(v,w) \neq 0$$ or $$\omega(v,w) = 0$$ actually tell us about the vectors $v,w$? ($\omega$ is the ...
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1answer
23 views

Planes And Lines

Given :Point $A(1,2,4)$ and plane $P: x-y+z+2=0$ How to find coordinates of point $A'$ the symmetric of point $A$ with respect to plane $P$
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0answers
13 views

An expression of covectors acting on vectors on the tangent space of a manifold

Let $M$ be a smooth manifold. Take $p\in M$ and $(U,\varphi)$, $\varphi:U\rightarrow \mathbb{R^n}$, a chart around $p$. Let $\mathbb{R}^n\left[\frac{\partial}{\partial x_i}\right]$ and ...
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0answers
39 views

Limit of the projection of a matrix when the projection is not continuous

Consider two real matrices: the $n\times n$ matrix $A$ the $n\times m$ matrix $B$ of rank $m$, with $m<n$. Let, for $a\in\mathbb{R}$, $$S_a=A-aI_n,$$ and denote by $P_a$ the orthogonal ...
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1answer
35 views

How to find the dimension of the given vector space

Let $L=\{p(B)|\ p\ \text{is a polynomial with real coefficients}\},$ where $B =\begin{pmatrix} 0 & 1 &0\\0 & 0&1\\ 1&0&0\end{pmatrix}.$ Then the dimension $\;d\;$ of the vector ...
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1answer
9 views

Considering a basis from two different space.

For a vector space V there is a orthonormal basis. If we watch these basis from a subspace of this vector space, are they still orthonormal?
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2answers
38 views

Infinite subspaces for a vector space that cannot be spanned by a single element

If a vector space (over an infinite field) cannot be spanned solely by a single element, does it mean it has infinite subspaces? I couldn't find an example that contradicts this
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1answer
11 views

Find parametric equations for the line through the point $(0,1,2)$ that is perpendicular to the line $x=1+t, y=1-t,z=2t$ and intersects this line.

My work so far: Since the lines are perpendicular, the dot product of their direction vectors should be $0$, so $<1,-1,2>\cdot <x,y,z>=0$. But I'm not sure where to go from here. I don't ...
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3answers
81 views

Then prove: $\vec{v} = \vec{0}$ if $\langle u,v\rangle = 0$

If $\vec{v} \in V$ such that $\langle u,v\rangle = 0$, $\forall \vec{u} \in V$. Then prove: $\vec{v} = \vec{0}$ I tired to solve by assuming that they are $\langle u,v\rangle \neq 0$ $\rightarrow$ ...
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2answers
52 views

What's the easiest way to find all $\alpha\in\mathbb{R}$ such that $\tiny\left(\begin{matrix}1&2\\2&\alpha\end{matrix}\right)$ is positive definite?

For which $\alpha\in\mathbb{R}$ is $$C:=\left(\begin{matrix}1&2\\2&\alpha\end{matrix}\right)$$ positive definite, positive semidefinite or indefinite? It seems to be a simple task, but for ...
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1answer
30 views

Find a plane that passes through the line $x-1=\frac{y-3}{-2}=z$ and is perpendicular to the plane $x+y-2z=1$

I'm mostly having trouble with the first part. How do I make sure the plane passes through the given line, $x-1=\frac{y-3}{-2}=z$? The second part seems easy enough; just set the dot product of the ...
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1answer
22 views

finite dimensional vector spaces of functions left invariant by translation

Let $E$ be a finite dimensional vector space of functions $\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall f \in E, \forall t \in \mathbb{R}, x \mapsto f(x-t) \in E$. Example of such spaces ...
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2answers
39 views

Prove $ |\langle u,v\rangle| = \lVert u \rVert \cdot \lVert v \rVert$

If V is the finite dimensional inner product space, then prove the following: If $u, v \in V$ are linearly dependent, then $ |\langle u,v\rangle| = \lVert u \rVert \cdot \lVert v \rVert$ Thanks.
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0answers
21 views

How to extended a unitary operator to a larger space?

Problem (the following is the exercise problem from Neilson and Chuang) Suppose $V$ is a Hilbert space with a subspace $W$. Suppose $U: W\rightarrow V$ is a linear operator which preserves inner ...
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0answers
33 views

Matrix integral $\int_{0}^{1}\exp\left(sA\right)BB^T\exp\left(sA^T\right)\,{\rm d}s$

Let $A\in M_{n}(\Bbb R)$ and $B\in M_{n,m}(\Bbb R)$ and $C=\int_{0}^{1}\exp\left(sA\right)BB^T\exp\left(sA^T\right)\,{\rm d}s$. Prove that $C$ is invertible if and only if $\sum_{i=0}^{n-1} ...
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3answers
26 views

Trying to figure out formula for deciding how to write Linear Transformation as a matrix relative to a basis

In these lecture notes: http://www.math.rice.edu/~hassett/teaching/221fall05/linalg5.pdf the formula (last line on first page) for finding a matrix relative to bases $B'$ and $B$ is: (1) $$ C_{B'}T ...
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3answers
35 views

Line of intersection of two planes

So, this question is more like two mini-questions that are subsets of a single regular-sized question. Say I have two planes: $x-z=1$ and $y+2z=3$. I'm trying to find their line of intersection. a. ...
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3answers
30 views

Help understanding Vector Space Axioms

I am having a difficulty trying to understand an axiom regarding vector spaces. There exists an element $0$ in $V$ such that $x + 0 = x$ for each $x\in \mathbb{R}$ Two examples, that I don't ...
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1answer
12 views

How to find representation of polynomial w.r.t different basis

Let $B$ be the basis of the vector space of polynomials of degree less than or equal to 2. $B = \{1, t-1,(t-1)^2\}$. Let $u = 2t^2-5t+6$. How do you find $u_b$, the coordinate vector of $u$ relative ...
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2answers
20 views

Obout the poset of subspaces of the vector space $\mathbb{R}$ over $\mathbb{Q}$.

Let $L$ the set of all subspaces of the vector space $\mathbb{R}$ over $\mathbb{Q}$, ordered by the set strict inclusion: $V_1<V_2$ iff $\{x\in V_1 \Rightarrow x \in V_2$ and there exists $y \in ...
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1answer
26 views

How do I find this basis given matrix representations?

Here is the question: Consider the multiplication operator $L_A:{\mathbb R}^2\to {\mathbb R}^2$ defined by $L_A(x)=Ax$ where $A=\left[\begin{array}{cc}2 &0\cr1 &-1\end{array}\right]$. Find an ...
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1answer
49 views

A problem of field in abstract algebra

If $V$ is a finite-dimensional vector space over the field $K$, and if $F$ is a subfield of $K$ such that $[K:F]$ is finite, show that $V$ is a finite-dimensional vector space over $F$ and that ...
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1answer
30 views

every finitely generated vector space has a basis. Question about the proof

Let $V$ be a finitely generated vector space over a field $K$. Then $V$ has a basis. I have a question about the proof we had in lecture. Proof: $V$ is finitely generated, this means for ...
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1answer
19 views

Scalar product with parameters

How do I get the values of the parameters in this equation? $\langle x,y\rangle = x_1y_1-2x_1y_2+ax_2y_1+bx_2y_2$ I do know that this equation shows a scalar product in $\mathbb{R}^2$, but how do I ...
2
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0answers
15 views

Relationships between affine closures and convex closures

Let $V$ denote a vector space. Then the following concepts make sense: affine subset of $V$ affine closure (affine "hull") of a subset of $V$ Suppose $V$ is in fact a real vector space. Then the ...
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1answer
17 views

Proving subset of vector space is closed under scalar multiplication

Let $V$ be the vector space of all continuous functions $f$ defined on $[0,1]$. Let $S$ be a subset of these functions such that $\int_0^1 f(x) = \int_0^1x f(x)$. To prove it is closed under scalar ...
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1answer
29 views

Prove $\dim(A) + \dim(B) = \dim(A+B)$ iff $A \cap B = \{0\}$ [duplicate]

$A,B$ are subspace of a finite-dimensional vector space $V$. Show that $\dim(A) + \dim(B) = \dim(A+B)$ if and only if $A \cap B = \{0\}$. It (kind of) seems intuitive but I'm having a hard time ...
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1answer
27 views

Find transformation matrix $T$ relative to new bases

T is a linear transformation represented as $\left(\begin{array}{ccc}1 & 1 & 0 \\0 & 2 & 0 \\3 & 1 & 0 \\0 & 1 & 1\end{array}\right)$ w.r.t the standard basis. Now ...
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1answer
31 views

Distance between function and subspace

Let $f(x)=cos^{n+1}(x)$, where $n \in \mathbb{N}$. In the real vector space $C([-\pi,\pi],\mathbb{R})$, we consider the inner product $\int_{\pi}^{\pi} \! f(x) g(x) dx$. My question is: What is ...
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1answer
52 views

About subspaces of $\mathbb{R}$ as vector space over $\mathbb{Q}$.

In many texts is noted the analogy between the transcendence degree of a field extension and the dimension of a vector space, so I'm tempting to use such analogy to better understand the structure of ...
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1answer
27 views

What do $F(-∞, ∞)$ and $C(-∞, ∞)$ stand for?

What do $F(-∞, ∞)$ and $C(-∞, ∞)$ stand for? They are vector spaces, with $C(-∞, ∞)$ being a subspace of $F(-∞, ∞)$. $C^1(-\infty, \infty)$ is a subspace of $C(-∞, ∞)$ and is defined as the set of ...
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1answer
22 views

Find basis for kernel and matrix representation

Problem 4 from https://math.berkeley.edu/~ogus/Math_54-07/Exams/midsol1.pdf $\beta$ is a basis of $P_3$, the set of all polynomials of at most degree 3.$\beta = (x^0,x^1,x^2,x^3)$. Let $T$ be a ...
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1answer
20 views

Write down basis for the set of all polynomials $f(x)$ of degree at most 5 such that $f(2) = 0$.

Write down basis for the set of all polynomials $f(x)$ of degree at most 5 such that $f(2) = 0$. I know there are lots of answers you could write, but would this be correct: $\{(x-2)^5, (x-2)^4, ...
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0answers
16 views

Vector dimensionality inference

If $\vec{u}$, $\vec{v}$ and $\vec{w}$ are vectors such that: $\vec{u}$ . $\vec{v}$ $\neq 0$ $\vec{u}$ . $\vec{w}$ $\neq 0$ $\vec{v}$ . $\vec{w}$ $\neq 0$ What can be said about the dimensionality ...
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1answer
24 views

If $\dim V = n$ and $S$ is a linearly independent set in $V$, then $S$ is a basis for $V$, True or False?

I'm currently taking a linear algebra course, and the topic of the current section is dimensions of vector spaces. I came across the titled question in the practice problems. Online sources that I've ...
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1answer
7 views

Vector Subspace involving polynomials

H={p(x)∈P2|p(1)=0} is a vector subspace of P2. What is a basis for for H and the dim(H)? I think the dimension is 0 since th restriction of p(1)=0, is that wrong because it is a polynomial?
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36 views

Algebraic extensions help?

$K$ is an extension field of $F$. If $[K : F]$ is finite and $u$ is algebraic over $K$, prove that $[F(u) : F]$ divides $[K(u) : F]$.
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1answer
27 views

Is the point on the left or the right of the vector in 2D space?

I'm trying to find if one point on the left or the right of a 2D vector. Example, looking to the figure below; I have the 2D points for a,b and c in the two cases. I'm try to find whether c is located ...
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2answers
17 views

How to find the orthogonal complement of a subspace?

I am having a hard time understanding how to find the orthogonal complement of a subspace $M$ of a vector space $V$. From my modest understanding, $M^\perp$ is a subspace of $V$ where all its ...
1
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0answers
34 views

Vector subspace. [closed]

$H = \lbrace p(x) \in P_2 \vert p(1) = 0 \rbrace $ is a vector subspace of $P_2$. What is a basis for for $H$ and the $\dim (H)$? I think the dimension is $0$ since th restriction of p(1)=0, is that ...
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2answers
40 views

Some question about extension of bounded linear operator

Let we have the following bounded linear operator $$T: D(T)\rightarrow Y$$ such that $D(T)$ is the domain and it is a vector space and $Y$ is a Banach space . Then it has an extension $$H: ...