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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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 this ...
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26 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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39 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
53 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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4answers
25 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
18 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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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 ...
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1answer
7 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
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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
24 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
20 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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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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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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1answer
16 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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1answer
32 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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0answers
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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
22 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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19 views

Which of the following is/are always correct. [on hold]

Let $A$ be a $4\times7$ real matrix and $B$ be a $7\times4$ real matrix such that $AB=I_4$, where $I_4$ is the $4\times4$ identity matrix. Which of the following are is/are always true ...
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0answers
11 views

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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2answers
70 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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1answer
15 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 ...
2
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0answers
23 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
28 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 ...
3
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0answers
55 views

Symmetric kernel of tensor product

Let $V,W$ be two vector spaces, and let $L_i:V\rightarrow W$, $i=1,\ldots,n$ be $n$ linear maps with disjoint kernels $K_i$ of dimension $1$. Consider the tensor product of these maps $L_1\otimes ...
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1answer
21 views

Domain of compostions of linear mappings [on hold]

Let $T$ be a linear transformation from $\Bbb R^3$ into $\Bbb R^2$ and $S$ be a linear transformation from $\Bbb R^2$ into $\Bbb R^3$. Is the mapping $ST$ a linear transformation from $\Bbb R^3$ into ...
2
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1answer
15 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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0answers
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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
16 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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2answers
50 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 ...
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3answers
35 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 ...
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3answers
54 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, ...
0
votes
1answer
18 views

Find vector and parametric vector of a line

I have a line that is perpendicular to a plane. This perpendicular line is $3i-2j+6k$. I've also been given that the line passes through $A(2,3,0)$. I'm unsure on how to represent this line as a ...
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3answers
27 views

Find the vector passing through a given point which is orthogonal to a given triangle in space

I'm given this problem where I have 3 points in space $A(3, -1, 2)$, $B(-2,1,2)$ and $C(2, 0, 5)$. I need to find the vector passing through point $A$ that is perpendicular to the triangle made by ...
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2answers
51 views

An example of non euclidean inner product [on hold]

Please give me an example of non euclidean inner product.Is there any method to construct such an inner product?
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1answer
22 views

Linear Algebra-Vector Subspaces Question [on hold]

Let $U=\{f\in C^1([-1,1],\Bbb R);y'=y+1\}$. Is $U$ a subspace of $C^1([-1,1],\Bbb R)$?
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Does $\dim (A_1\otimes A_2)=\dim(V_1\otimes V_2)$ for all affine spaces $A_{1,2}$, their vector spaces $V_{1,2}$ and the operations $\cap,+$?

Let $A_1=P_1+V_1,A_2=P_2+V_2$ be affine spaces. My teacher uses $\dim$ on affine spaces and the embedded vector spaces interchangeably, which is correct by definition for $\dim A_1=\dim V_1$, but ...
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0answers
11 views

Show that W is a Subspace of R³

can you help me: Let $u=(1,2,-3)$ and $v=(-2,3,0)$ Two Vectors in R³ and let W the subpace of R³ that consists of all the vectors shape $au+bv$, where, $a,b ∈ R $ show that W is subspace of R³ im ...
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1answer
23 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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1answer
27 views

Connection between algebraic multiplicity and dimension of generalized eigenspace

Assume $V$ to be a finite dimensional vector space. Define the algebraic multiplicity $am(\lambda)$of an eigenvalue $\lambda$ of a linear operator $T:V\to V$ as the maximum index of the factor ...
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How to show if 2 vectors are the same

I've been given 2 lines in different forms $L1$ is $$\frac{x-1}{4} = \frac{y-2}{3} = \frac{z-10}{5} $$ $L2$ is $$x = -7-4t$$ $$y = -4-3t$$ $$z = -5t$$ I've converted $L2$ into its Cartesian form as ...
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1answer
28 views

How many elements are there in the vector space over F of dimension 5

When $F = \mathbb Z_2$ (the two element field), how many elements are there in the vector space over $F$ of dimension $5$? Would it be $32$? Thank you so much.
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Is the Laplace transform a vector space isomorphism? And what space is it isomorphic to?

The laplace transform is a linear transformation, $\mathcal{L}: \mathcal{M} \rightarrow?$, where $\mathcal{M}$ is the set of exponentially bounded functions on $\mathbb{R},$since ...
2
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2answers
23 views

Basis for the vector space P2

I am trying to wrap my head around vector spaces of polynomials in P2. If I represent the polynomial $ ax^2 + bx + c $ with the matrix $ A = \begin{bmatrix} 1,0,0 \\ 0,1,0 \\ 0,0,1 \\ \end{bmatrix} ...
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1answer
48 views

Show that $\| u - v \|^2 = \| u - P_U(v) \|^2 + \| v - P_U(v) \|^2 $ and minimize $d(u, v)$

i) Let $\left(V, \langle\ ,\ \rangle\right)$ be an inner-product space, $v \in V$, and let $U$ be a subspace of $V$ with the orthogonal projection map $P_U$. Show that $ \| u - v \|^2 = \| u - P_U(v) ...
2
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1answer
37 views

calculating characteristic polynomial in $\mathbb{R}^n$

Given some hyperplane arrangement $\mathcal{A}$, we call any subset $\mathcal{B}\subseteq \mathcal{A}$ $\textit{central}$ if $$\displaystyle \bigcap_{H\in \mathcal{B}}H\neq \emptyset.$$ There is a ...
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3answers
44 views

Linear Algebra, kernel [on hold]

Suppose that $W$ and $V$ are vector spaces, and that $f : V \mapsto W $ 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 ...
2
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4answers
46 views

How to test if these are a vector space and find the basis?

I have been trying to work through these linear algebra questions in my text book for hours now, but i just cant seem to figure it out. The question is: ...
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1answer
26 views

Proof for the necessity of conditions for a subspace

In [Axler 2015], Theorem 1.34 states that A subset $U$ of $V$ is a subspace of $V$ if and only if $U$ satisfies the following three conditions: additive identity: $0\in U$; closed ...
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0answers
20 views

Divergence of scalar field and curl of scalar field?

I have V(x.y)=(y^c,x^c) for positive c and r(x.y)=(x.y) I want to find div(V X r) and Curl (V X r). So V X r is determinant and it is scalar field. I got f(x.y)=y^(c+1)-x^(c+1) Thus, div(V x r) = ...
0
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1answer
17 views

Finding a vector equation for a trajectory

A shell is fired from the ground with muzzle speed of 320 ft/s and elevation angle of 60 degrees (assume $g=32 \, \mbox{ft}/\mbox{s}^2$) Find a vector equation for the shell's trajectory ...