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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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 ...
7
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3answers
74 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 $\langle u,v\rangle \neq 0$ $\rightarrow$ ...
2
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2answers
50 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 ...
0
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1answer
25 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 ...
1
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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 ...
2
votes
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.
2
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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
31 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
23 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 ...
0
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3answers
33 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. ...
1
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3answers
27 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 ...
0
votes
1answer
11 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 ...
1
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2answers
18 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 ...
0
votes
1answer
24 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 ...
1
vote
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 ...
2
votes
1answer
28 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 ...
1
vote
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
votes
0answers
12 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 ...
0
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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 ...
-2
votes
1answer
26 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 ...
0
votes
1answer
25 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 ...
1
vote
1answer
29 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 ...
1
vote
1answer
51 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 ...
0
votes
1answer
26 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 ...
0
votes
1answer
21 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 ...
1
vote
1answer
19 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, ...
1
vote
0answers
14 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 ...
0
votes
1answer
22 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 ...
0
votes
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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0answers
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 $[K(u) : F]$.
1
vote
1answer
25 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 ...
0
votes
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
vote
0answers
33 views

Vector subspace. [on hold]

$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 ...
-1
votes
2answers
38 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: ...
0
votes
0answers
19 views

Prove that there exits an automorphism from $G$ to $G$ when dim G=infinite

Suppose $G$ is a vector space over $\mathbb Z_2$ . The problem is to prove that there exits an automorphism from $G$ to $G$ Now $G$ has a basis say $\{b_1,b_2,...,b_n\}$.Then any $g\in G$ can be ...
0
votes
1answer
40 views

Determining diagonalizability of a matrix containing complex enteries

$$A=\left[\begin{matrix}3-8i&-11+7i\\-1-4i&-2+6i\end{matrix}\right]$$ I've determined the $tr(A) = 1-2i$, and the $det(A)=3-3i$. From here I should be able to use the characteristic equation ...
0
votes
1answer
31 views

Finding linear dependance of a set of functions

where the set $B = \{1+2x+2x^2-x^3,3+2x+x^2+x^3,2x^2+2x^3\}$, how can I show they are linearly independent? Could I set the three vectors, u, v, w, into a coefficient matrix and find it's ...
0
votes
2answers
20 views

How to show a set of vectors are a basic for a given plane

To determine if a set, B, of the vectors, u, and v for a basis for the plane, W. let u=(1,2,-1), and v=(1,1,1), W =-3x+2y+z=0 I was able to determine the two vectors, w[1], and w[2], from s and t ...
-1
votes
1answer
12 views

Adding Vectors With A Missing Angle or Magnitude [on hold]

The following introduction problem is as given: An airplane has a bearing of 300° at a speed of 400 mph. The airplane encounters wind of velocity 75 mph in the direction N 40° E (60°). Find the ...
0
votes
0answers
18 views

Find the possible signatures of the bilinear forms

Find the possible signatures of the following bilinear forms: The bilinear form $\phi:\mathbb R^n\times\mathbb R^n\to\mathbb R$ given by $\phi(x,y)=x^Tp(A)y$ where $p(t)=t^2+bt+c$ is a ...
1
vote
1answer
17 views

If $H$ is a bilinear form then for every $x$ there exists non-null $y$ with $H(x,y)=0$

Prove or disprove: Suppose $H$ is a bilinear form on a finite dimensional vector space $V$, with $\dim(V)>1$. Then for any $x\in V$ there always exists a non-zero $y\in V$ such that $H(x,y)=0$. ...
1
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0answers
20 views

How to solve matrix differential equations problem

I need to solve $$\begin{bmatrix} 20 & 6 \\ 6 & 7 \end{bmatrix} \begin{bmatrix} y''_1 \\ y''_2 \end{bmatrix} = \begin{bmatrix} 40 \\ 15 \end{bmatrix}$$ And I'm ...
5
votes
1answer
52 views

Prove that the isomorphism between vector spaces and their duals is not natural [duplicate]

In preparation for an introductory talk on category theory, I recently spent some time thinking about natural transformations. The first example, or maybe the second, that everyone gives to motivate ...
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0answers
10 views

graph has no bridge iff a spanning subgraph of the graph is the support of a flow

A $\textit{bridge}$ of a graph $G=(V,E)$ (finite graph and we allow loops and multiple edges) is an edge $e$ whose removal disconnects $G$. Let $\mathcal{O}$ be an orientation of the edges of $G$. ...
0
votes
1answer
22 views

How do I change this basis for a transformation?

I have $$\left[ L\right]_\mathcal{B}^\mathcal{B} = \begin{pmatrix}2&2&-1\\7&4&-2\\8&5&2\end{pmatrix}$$ and I want to get $[L]_\mathcal{E}^\mathcal{E}$ where the ...
-1
votes
0answers
25 views

3D vector perpendicular calculation

Three points $A(6,7,-6)$,$ B(0,0,0)$ and $C(2,6,9)$ are given which are the vertices of a cubes. Find the coordinates of another vertex not on the $ABCD$ plane. I found the answer by finding the ...
0
votes
1answer
47 views

Computing the dimension of a vector space in terms of matrix rank

Let $V=\mathbb C^n$ be a complex vector space, and $A,B:V\to V$ two commuting endomorphisms. I am interested in determining the dimension of the vector space $$F_{AB}=\{(a,b)\in V\times V\,|\,A\cdot ...
1
vote
1answer
33 views

linear algebra unclear terminology on direct sum

I was given this question in a linear algebra assignment It tells us that V is a vector space over an infinite field F and $ W \subset V $ is a non trivial subspace of V (neither $ V $ nor the the ...
-3
votes
0answers
40 views

Vector Space, Basis, Matrices [closed]

Let $S_n$ denote the vector space of all $n\times n$ symmetric matrices (i.e. $M = M^T$). Let $A_n$ denote the vector space of all $n \times n$ anti-symmetric matrices (i.e. $M^T = - M$). (a) Find a ...
-1
votes
0answers
17 views

Find u-v with given magnitudes [closed]

Vector u has a magnitude of 9 and vector v has a magnitude of 6. The questions asks to find the magnitude of vector (u-v). I am unsure of how to approach this problem because I thought you would ...