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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51 views

Finding a pair of Orthogonal Vectors

Want: Pair of orthogonal vectors in $R^4$ that are also orthogonal to the vector (1,1,-2,3) My attempt at a solution: I got stuck...
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1answer
25 views

Let W be the collection of all 2 by 2 symmetric matrices. Describe the orthogonal complement of W. (please)

A matrix is symmetric if $A^T$=A And the standard basis for symmetric matrices is [a,b], [b c] written as rows of a 2x2 matrix (sorry don't know how to make a matrix on this site). My question: How ...
3
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1answer
65 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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31 views

Vector Space Basis' Proof

Show that : if $B=\{X_1, X_2, \ldots, X_n\}$ and $A= \{ Y_1, Y_2, \ldots, Y_p\}$ are basis' of a vector space $(E, +, \cdot)$ that means $n=p$. I have no idea on how to start this proof, if I can get ...
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2answers
26 views

How is a Euclidean space a function space?

To be more precise, in what sense is $\mathbb R^N$ a function space? I quote from page number 3, in the first chapter of "Introduction to Hilbert Spaces with Applications" by Debnath and Mikusinski ...
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19 views

Let $M =\{ f(x)\in P_3 | \int_0^1f(x)dx = 0\}$ Find basis for M.

Let $M =\{ f(x)\in P_3 | \int_0^1f(x)dx = 0\}$ Find basis for M. solution: $P_3$ is the set of all polynomials of degree strictly less than 3, ($f(x) = a_2x^2+a_1x+a_0$). hence, $\int_0^1f(x)dx = ...
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Prove $K\cap L$ is a subspace of V, but $K\cup L$ is never a subspace.

assume K, L are proper subspaces. Prove $K\cap L$ is a subspace of V, but $K\cup L$ is never a subspace. Solution: if $v_1,v_2\in K$, then $c_1v_1+c_2v_2 \ in K$ [because K is a subspace] if ...
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2answers
35 views

Prove that if v is orthogonal to u, then it is orthogonal to any scalar multiple of u.

I never understand where to start with proofs, but whenever I see them done I understand them. My attempt: For this one could I just use the property of inner products to prove this? That being ...
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2answers
33 views

Prove that if u and v are vectors in $\mathbb{R}^n$, then $\langle u,v\rangle =1/4\|u+v\|^2-(1/4\|u-v\|^2)$

I seem to always have troubles when starting proofs. My professor said that the proofs he gave us today are mostly one line proofs, but I just don't know where to start with this one. What I've ...
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2answers
31 views

Vector Spaces and Linear Transformations ($T^2 = 0 \iff R(T) \subseteq N(T)$). [duplicate]

Let $V$ be a vector space over a field $F$. Let $T: V\to W$ be a linear transformation. a. Prove that $T^2=0$ if and only if $R(T)$ is contained in $N(T)$. (Here we denote $T^2$ as the linear ...
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3answers
38 views

find dimension of a vector space

Let A $\begin{pmatrix} 1 & 2 & -1\\ -2 & -4 & 2\\ 0 & 1 & 2\\ \end{pmatrix} $ . Let D = $\{B\in\mathbb{R}^{3x3}| BA = \begin{pmatrix}0 &0&0\\0 &0&0\\0 ...
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2answers
19 views

how to show $F=\{(a+2b+3c,a-b,-3a+b-2c,2b+2c),\}$ is a subspace?

how to show $F=\{(a+2b+3c,a-b,-3a+b-2c,2b+2c),\}$ is a subspace? i understand closed subspace should be closed under addition and scalar multiplication
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1answer
35 views

How do you find a basis for $\mathbb R^4$ such that it contains specific elements

How do you find a basis for $\mathbb R^4$ such that it contains specific elements: $(2,4,-1,0), (-4,-8,2,1)$
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2answers
15 views

Dimension of an $F$-vector space, compared to dimension over $E:F$

Suppose $E:F$ is a finite field extension of $F$. If $V$ is both an $F$-vector space and an $E$-vector space, then is there any relation between the dimension of $V$ over $F$ and the dimension of $V$ ...
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1answer
12 views

Invariant subspace (Proof)

How do I prove, that the eigenspaces of $T^n$ are invariant in regard to $T$, assuming T is an endomorphism in a real vector space V $(T: V\rightarrow V)$? That's how I started: Let $E_\lambda$ be ...
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2answers
35 views

Determining $\dim V$

This is probably a simple question but I'm not sure about the answer. According to Lounesto in Clifford Algebras and Spinors, a vector space of $\dim V=2$ has basis such as \begin{align} ...
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2answers
32 views

Correspondence between linear maps of a vector space into itself and linear maps of the dual into itself.

I was wondering about vector spaces and their dual. Specifically, in the context of finite-dimensional vector spaces, I asked myself if it is true that there is a one-to-one correspondence between the ...
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2answers
42 views

How do I find a relation for these polynomials from a matrix?

I have the following three polynomials: $1 + 2t^2, 4 + t + 5t^2, 3 + 2t$. I need to show that they are linearly dependent in $\mathbb P_2$ (polynomials of degree at most $2$). I put them in a $3x3$ ...
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1answer
30 views

Suppose $S_1 =\{ u_1 , u_2 \}$ and $S_2 = \{ v_1 , v_2 \}$ are each independent sets of vectors in an n-dimensional vector space V.. [duplicate]

Let us assume that every vector in S_2 is a linear combination of vectors in S_1. Question: Does that mean that S_1 and S_2 are bases for the same subspace of V? I know that the answer to this ...
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0answers
35 views

show linear transformation bijective

Can you please help me prove this? Let $T:\mathbb{R}^7\to\mathbb{R}^7$ be a linear transformation such that 9 is an eigenvalue of $T$ and $dim(E_9)=6$ Prove that either T-4I or T-5I is a bijection ...
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46 views

Suppose $S_1 =\{ u_1 , u_2 \}$ and $S_2 = \{ v_1 , v_2 \}$ are each independent sets of vectors in an n-dimensional vector space V.

Let us assume that every vector in $S_2$ is a linear combination of vectors in $S_1$. Question: Does that mean that $S_1$ and $S_2$ are bases for the same subspace of $V$? I know that the answer to ...
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3answers
61 views

Determine whether the set $\{v_1 + v_2 - v_3, 2v_1 + 2v_3, -v_1 + v_2 - 3v_3\}$ is linearly dependent or independent.

We had a question on our last test that was very similar to this and I only got $2$ points of $6$ and I want to make sure I do it right this time. Here's my solution to that one: Let $v_1, v_2,$ and ...
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2answers
30 views

Proving kerT is a subspace of V. and rangeT is a subspace of W.

My question is as follows: Suppose $V$ and $W$ are vector spaces, and let $T: V \longrightarrow W$ be a linear transformation. Show that $\ker T$ is a subspace of $V$. Show that ...
5
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1answer
193 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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4answers
39 views

Let V be a vector space and W a subset of V. Suppose zero is in W and W is closed under addition. Is W a subspace of V?

I know that the answer to this question is No. My question is why is the answer no? What's missing? if possible give a specific example of both V and W such that W satisfies above conditoins but it ...
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1answer
20 views

How do I find a basis for the following subspace?

I'm unsure how to do the following problem: Find a basis of the following subspace of $R^4$. W = all vectors of the form $(a,b,c,d)$ where $a+b-c+d=0$. Any help would be great, many thanks :)
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1answer
17 views

Determine rank and nullity of linear transformation between polynomial of degree $\leq$ 5 to $R^6$

Define the mapping $T$to be the one that maps a polynomial $f(x) \in V$ to the vector $(f(0), f(1),f(2),f(3),f(4),f(5))^t$, where $V$ is the vector space of all real polynomials of degree 5 or less. ...
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1answer
30 views

Not understanding what linear groups are, please need help on the questions 1-4

Above is my math homework. I am in a linear algebra class that is the first linear algebra course i am taken and am overwhelmed with the problem. I am not understanding what to do, but i understand ...
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1answer
28 views

Is Basis of a vector space a subset of the vector space

Now, I was going through my notes which says that basis of a vector space V is a set S such that 1)S is a linearly independent set 2)v=L(S) Now there might be multiple basis of a vector space.Hence ...
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2answers
36 views

point of intersection of a line l and the plane p, i get 0 somehow

$L: x=\frac {y-1}{2}=\frac {z+1}{3}$, $P= x − 2y + z = 1$. Find the point of intersection of the line L and the plane P.
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2answers
33 views

Let $T\colon V\to V$ be a linear transformation such that $\dim(V)=n<\infty$. Prove that $T$ is bijection >iff T is injective.

Let $T\colon V\to V$ be a linear transformation such that $\dim(V)=n<\infty$. Prove that (a)$T$ is bijection iff (b)T is injective. Solution: show $(a)\implies(b)$ If $T$ ...
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2answers
32 views

vectors and cartesian equation on the line in 3d

Find in scalar parametric form an equation for the line of intersection of the plane $P$ and the plane with Cartesian equation $2x + y − z = 0$. $P= x − 2y + z = 1$.
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1answer
22 views

Divergence of $\phi$ from p

I am reading a paper which is based mostly on divergence. I tried to get a basic understanding of divergence but I cannot see how it is linked with this aspect. It says: $D(\phi,p) = \phi . ...
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2answers
73 views

Find vectors when added up equal (1, 1, 1)

Question: Let $V$ be the 2-dim subspace of $\mathbb R^3$ spanned by $(1, 2, -3)$ and $(-2, 0, 1)$. Write the vector $u = (1,1,1)$ in the form $u = v + w$, where $v$ is in $V$ and $w$ is in $V^\perp$, ...
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0answers
15 views

Write F+K as a span of some basis.

Let $F = \{(a,a+b,4b,0) | a,b\in \mathbb{R}\}$ and $K = \{ (c,2c+d,4c-d,2d) | c,d\in \mathbb{R}\}$. Write F+K as a span of some basis. Solution: The basis of $F$ is $\{(1,1,0,0),(0,1,4,0)\}$. ...
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1answer
37 views

I need help with this linear transformation.

Please let me know if my process or thinking is incorrect at any point. Let $T:P_3 \rightarrow P_3$ be the linear transformation such that $$T(-2 x^2)= 3 x^2 + 3 x,\\T(0.5 x + 4)= -2 x^2 - 2 x - 3, ...
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1answer
27 views

Prove that for every $k$ there's an invariant subspace

Let $V$, a vector space above $\mathbb{C}$ and let $T:V\to V$, a linear transformation. Show that for every $0\le k \le n$ there is an invariant subspace of $T$ with a dimension $k$. It seems ...
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1answer
19 views

Green's theorem in divergence form and its line integral?

$$ \int_C F \times da $$ $$ k\iint_R \operatorname{div} F \ dx \, dy $$ Hi Let $F$ be two-dimensional vector field. State a definition for the vector-valued line integral so that your definition ...
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How can I do this line integral using stoke's theorem??

$$ \int_{C}{(y^2+z^2)dx}+(x^2+z^2)dy+(x^2+y^2)dz $$ where C is the intersection of hemisphere $x^2 + y^2 + z^2 = 2ax, z \geq 0$ and $x^2 + y^2=2bx $ where 0 < b < a. Compute line integral ...
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0answers
28 views

computing characteristic polynomial of hyperplane arrangement

The following problem comes from Richard Stanley's $\textit{Enumerative Combinatorics}$ vol. 1, 2nd ed. It is problem 114 (c) in Chapter 3. Let $\mathcal{A}$ be a hyperplane arrangement in ...
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1answer
52 views

Give an example of three different points in $\mathbb R^3$ such that there are infinitely many planes in $\mathbb R^3$ passing through all of them.

A past exam question. I'm not certain on the meaning. I assume it wants a $3$ points on a straight line, one which case there would be infinitely many planes passing through all of them. But that ...
2
votes
1answer
335 views

Find normal vector of circle in 3D space given circle size and a single perspective

I don't really know what to search up to answer my question. I tried such things as "ellipse matching" and "3d circle orientation" (and others) but I can't really find much. But anyways... I have ...
2
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1answer
21 views

Does a vector have to be continuous to fall within a set?

The question asks: explain why $\ f(x) = $ $\ x \over \ x^2 + 4x + 3$ is a vector in $C[0, 3]$ but not a vector in $C[-3, 0]$. I know that $f$ is not continuous on $C[-3, 0]$ at $x = -1$ and $x = 3$. ...
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2answers
351 views

proof of the full exchange lemma

Let V be spanned by $\{v_1,...,v_k\}$ and let $\{u_1,...,u_k\}$ be a linearly independent subset of V, then: 1) $k\leq n$ 2) $\exists$ a spanning set $\{w_1,...,w_n\}$ for V where $w_i = u_i$ for $1 ...
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0answers
19 views

A thought about transition matricies in vector spaces

I am trying to work out the relationship between transformation matricies of a vector space with different bases. I came up with an equation which does not look right, but I would like your opinion. ...
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1answer
17 views

functions and the commutative property

with regard to vector spaces of functions. How do I know if the commutative property holds for a set of functions. especially if the vector space includes an infinite set. for instance, for the ...
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3answers
71 views

For which values of a do the following vectors form a linearly independent set in R^3

I've seen this same question, but asking for linearly dependent, not linearly independent. $$ V_1= \left(a,\, \frac{-1}{2}, \,\frac{-1}{2}\right),\;\; V_2= \left(\frac{-1}{2},\, a, ...
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5answers
123 views

Proving rank of $AB$ is at most equal to rank of $B$

$A=m\times n$ matrix. $B = n\times p$ matrix. Prove that the rank of of the product $AB$ is at most equal to the rank of $B$. Current state of my work: (1) First idea: show that the kernel of $B$, ...
0
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3answers
49 views

Proof of linear independence of non-empty subsets

The question states: Show that if $S = \{v_1, v_2, \ldots , v_r\}$ is a linearly independent set of vectors, then so is every non-empty subset of $S$. I understand that if $r>n$, $S$ is ...
0
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2answers
62 views

Vector spaces whose elements are functions

I'm trying to understand what a vector of functions is, from trying to understand how to solve linear homogeneous differential equations. It seems that functions can be manipulated as vectors as ...