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1) Base case ##k=2##
##U_1 + U_2 = span (U_1 \cup U_2)##, which I understand how to prove is OK.
2) Induction hypothesis
We assume that the following statement holds
$$U_1 + U_2 + ... + U_{k-1} = span (U_1 \cup U_2 \cup \ ... \ \cup U_{k-1})$$
3) Induction step
$$U_1 + U_2 + ... + U_k =...
I have a weight of 30,000 lbs distributed across a span of 100' and supported at the ends. If I have 3 beams to support this load, each beam would then need to hold 10,000 lbs, and with a F.S of 1.6 each beam would need to hold 16,000 lbs. If I assume my max deflection at 1", then I can find the...
$\tiny{311.1.7.9}$
For what values of $h$ is $v_3$ in Span $\{v_1,v_2,v_3\}$ linearly \textit{dependent}. Justify
$v_1=\left[\begin{array}{rrrrrr}1\\-3\\2\end{array}\right],
v_2=\left[\begin{array}{rrrrrr}-3\\9\\-6\end{array}\right],
v_3=\left[\begin{array}{rrrrrr}5\\-7\\h\end{array}\right]$...
I have a trouble showing proofs for matrix problems. I would like to know how
A is invertible -> det(A) not 0 -> A is linearly independent -> Column of A spans the matrix
holds for square matrix A. It would be great if you can show how one leads to another with examples! :)
Thanks for helping...
Hey! :o
Let $1\leq n,k\in \mathbb{N}$ and let $v_1, \ldots , v_k\in \mathbb{R}^k$. Show that:
Let $w\in \text{Lin}(v_1, \ldots , v_k)$. Then it holds that $\text{Lin}(v_1, \ldots , v_k)=\text{Lin}(v_1, \ldots , v_k,w)$.
Let $v_1, \ldots , v_k$ be linearly dependent. Thn there is a $1\leq...
If ##\textbf{u}_1,...,\textbf{u}_n## form a basis in a linear space, how does one determine the dimension of the span ##\textbf{u}_1-\textbf{u}_2, \textbf{u}_2-\textbf{u}_3,...,\textbf{u}_n-\textbf{u}_1##? Since ##\textbf{u}_1,...,\textbf{u}_n## form a basis, they're linearly independent. If one...
Determine if the polynomial
$3x^2+2x-1$
is the $\textbf{span}\{x^2+x-1,x^2-x+2,1\}$ok from examples it looks like we see if there are scalars such that
$c_1(x^2+x-1)+c_2(x^2-x+2,1)=3x^2+2x-1$so far not sure how this is turned into a simultaneous eq
I did notice that it is common to get...
Determine if $A=\begin{bmatrix}
1\\3\\2
\end{bmatrix}$ is in the span $B=\left\{\begin{bmatrix}
2\\1\\0
\end{bmatrix}
\cdot
\begin{bmatrix}
1\\1\\1
\end{bmatrix}\right\}$
ok I added A and B to this for the OP
but from examples it looks like this can be answered by scalors so if...
Hello!
As the topic suggests I´m interested which functions space square waves span?
Lets say we define them as https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8953debf86627276f45bf8822140ff2bbaee56 .
Do they span the same space as the sines and cosines in Fourier analysis? :/
Thanks!
Homework Statement
I know how to approach this problem; however, I'm just confused as to why we consider that R^2 is a vector space over the field R, and not Q or any other field for this question?
Standard basis vectors: e_1, e_2 or i,j
Hello smart physicists!
I'm hoping to run a span of 124" for a pull up bar, with a support in the middle at the 62" point (so 3 supports total). I'm using the Rogue P-5V Pull-Up Bar Brackets (https://www.roguefitness.com/rogue-p-5v-garage-pullup-system). The brackets call for a tube with a...
Hey! :o
We have the subset $X_i$ of $\mathbb{R}^2$:
$$X_1 := \{(x,y) \in \mathbb{R}^2 : x + y = 0\}; \\ X_2 := \{(x,y) \in \mathbb{R}^2 : x + y = 1\} \\
X3 := \{(x,y) \in\mathbb{R}^2 : x^2 + y^2 = 0\}; \\ X4 := \{(x,y) \in \mathbb{R}^2 : x^2- y^2 = 0\}$$ We want to check which of these sets...
I'm trying to figure out something regarding the span of a set of vectors. Say we have a set of vectors ##V=\{v_1, v_2, ... v_k\}## in ℝn. If k < n then the set does not span ℝn. Why is this? Are there vectors in ℝn that aren't combinations of the vectors in ##V##?
Homework Statement
Let ##S_1## and ##S_2## be subsets of the vector space ##V##. Prove that ##span (S_1 \cap S_2) \subseteq span(S_1) \cap span(S_2)##. Given an examples of ##S_1## and ##S_2## for which equality holds and for which the inequality is strict.
Homework EquationsThe Attempt at a...
Hello all and thanks in advance for the help!
I am currently working on creating a lighting fixture that will be suspended over an aquarium. What I am having trouble with is determining a safe material and thickness for the spars going between the lights (see below). My goal is to have these...
If I construct a set of qubit gates, say {G1, G2 ... Gk ... Gn}, that can act on a state |ψ>, what does it mean for the set of states Gk |ψ> to span the Bloch sphere?
As an example, take the set {G1, G2, G3, G4} = { I, X π/2 , Y π/2, Xπ }
Here, X π/2 denotes a π/2 rotation about the x-axis, Y...
Hello everyone, I'm a CS student and I'm taking a course called Linear Algebra
it's very easy, but there is one thing that I'm not clearly understanding
i know how the general way to prove if given vectors span a vspace,
ex : v1,v2,v3 i put them in a Matrix form and prove the determinant...
Homework Statement
In this question , is figure 11-2 a and 11-2 b the same beam ? Why in a , the support is pinned end , it change to fixed end in figure b ?
Homework EquationsThe Attempt at a Solution
In equation 11-9 , i gtet different ans with the author ..
Here's what i gt :
MN = 2EK( 2...
I just read something that I do not want to misinterpret.
If there are two orthonormal basis that span the same space, which I think implies that each basis can be written in terms of the other basis, then measurements made with respect to each basis will not commute?
Does this mean that...
##\def \sqx{\frac{1}{\sqrt{2}}}##
If Szu represents spin-up in the z-direction and Szd represents spin-down in the z-direction then the vector which represents spin-up in the x-direction is given by the superposition of the z states \begin{equation}|S_{zu}\rangle =\begin{bmatrix}1 \\...
Homework Statement
The question is:
if vectors v1, v2, v3 belong to a vector space V does it follow that:
span (v1, v2, v3) = V
span (v1, v2, v3) is a subset of V.[/B]
2. The attempt at a solution:
If I understand it correctly the answer to both questions is yes.
The first: the linear...
An orthogonal basis set spanning R4 has four vectors, v1, v2, v3 and v4.
If v1 and v2 are
[ −1 2 3 0 ] and [−1 1 −1 0 ]
ﬁnd v3 and v4.
Please explain this in a very simple way.
As the title says, why does the set of hydrogen bound states form an orthonormal basis? This is clearly not true in general since some potentials (such as the finite square well and reversed gaussian) only admit a finite number of bound states.
Find a set of vectors {u, v} in $\mathbb{R}^4$ that spans the solution set of the equations:
$x - y + 2z - 2w = 0$
$2x + 2y -z + 3w = 0$
($u$ and $v$ are both $4 \times 1$)
$u = ?$, $v = ?$
I put the matrix in RREF to get
$\begin{bmatrix}1&0&3/4&-1/4\\0&1&-5/4&7/4\end{bmatrix} =...
This question was inspired by 3c) on https://people.phys.ethz.ch/~muellrom/qm1_2012/Solutions4.pdf
Given the operator
\hat{B} = \left(\matrix{b&0&0\\0&0&-ib\\0&ib&0}\right)
I find correctly that the eigenvalues are \lambda = b, \pm b.
To find the eigenvectors for b, I do the following...
Homework Statement
Please see the attached picture
Homework Equations
Reduced echelon form of the column matrix
The Attempt at a Solution
I can solve for the first part to find which ones are the bases in ##\mathbb{R}^3## by determining whether in the echelon form, there is a pivot in each...
I'm given the example that the space $\mathbb{R}^2$ is spanned by each of the following set of vectors: \left\{i, j\right\}, \left\{i, j, i+j\right\}, and \left\{0, i, -i, -j, i+j\right\}.
However, it's not obvious to me how. Let $i = (s, t)$ and $j= (u, v)$ then $\left\{i, j\right\}$ means...
Homework Statement
Determine whether the set spans ℜ3. If the set does not span ℜ3 give a geometric description of the subspace it does span.
s = {(1, 0, 3), (2, 0, -1), (4, 0, 5), (2, 0, 6)}
Homework EquationsThe Attempt at a Solution
I am having trouble with the second part of this problem...
Homework Statement
Let U is the set of all commuting matrices with matrix A= \begin{bmatrix}
2 & 0 & 1 \\
0 & 1 & 1 \\
3 & 0 & 4 \\
\end{bmatrix}. Prove that U is the subspace of \mathbb{M_{3\times 3}} (space of matrices 3\times 3). Check if it contains span\{I,A,A^2,...\}. Find the...
Homework Statement
Find the span of U=\{2,\cos x,\sin x:x\in\mathbb{R}\} (U is the subset of a space of real functions) and V=\{(a,b,b,...,b),(b,a,b,...,b),...,(b,b,b,...,a): a,b\in \mathbb{R},V\subset \mathbb{R^n},n\in\mathbb{N}\}
Homework Equations
- Span
-Subset
The Attempt at a Solution...
Homework Statement
Find the span of U=\{2,\cos x,\sin x:x\in\mathbb{R}\} (U is the subset of a space of real functions) and V=\{(a,b,b,...,b),(b,a,b,...,b),...,(b,b,b,...,a): a,b\in \mathbb{R},V\subset \mathbb{R^n},n\in\mathbb{N}\}
Homework Equations
-Vector space span
-Linear independence...
My linear algebra is a bit rusty.
Let ##A=\{\bar{v}_1, \dots, \bar{v}_1\}## be a set of vectors in ##R^n##. Can dim(span##(A))=n## without spanning ##R^n##?
I guess I'm unclear on how to interpret the dimension of the span of a set of vectors.
I'm stuck on a relation issue if there is a direct relation at all.
If I were to verify that a subset is a subspace of a vector space V, would it then be correct to check that subset for linear independence to verify that the subset spans the subspace? I'm not sure if I'm following the...
Homework Statement
Consider the vectors ##\vec{v_{1}},\vec{v_{2}},...,\vec{v_{m}}## in ##\mathbb{R}^{n}##. Is span ##(\vec{v_{1}},...,\vec{v_{m}})## necessarily a subspace of ##\mathbb{R}^{n}##? Justify your answer.
Homework EquationsThe Attempt at a Solution
I understand the three conditions...
So that's the question in the text.
I having some issues I think with actually just comprehending what the question is asking me for.
The texts answer is: all 3x3 matrices.
My answer and reasoning is:
the basis of the subspace of all rank 1 matrices is made up of the basis elements...
Given the following diagram:
The conductor(C) of known volume(##V##) and resistance(##R##), passes a uniform magnetic field(##B##)at a constant velocity(##v##), via the following formula: ## \epsilon = -vBL## the induced EMF can be calculated. This conductor acts as a power source to a certain...
Hi all,
This isn't actually part of my assigned homework, I was just trying it out as the topic confuses me. I think I might understand what's going on a little more if someone could walk me through this. Any advice on the intuition behind it would be great. Thanks so much.
1. Homework...
Homework Statement
A. Let {t,u,v,w} be a basis for a vector space V. Find dim(U) where
U = span{t+2u+v+w, t+3u+v+2w, 3t+4u+2v, 3t+5u+2v+w}
B. Compute the dimension of the vector subspace V= span{(-1,2,3,0),(5,4,3,0),(3,1,1,0)} of R^4Homework EquationsThe Attempt at a Solution
I know that...
Homework Statement
Let A be an m \hspace{1 mm} x \hspace{1 mm} n matrix, and let \vec{b} be a vector in \mathbb{R}^{m} . Suppose that \vec{b} is a linear combination of the columns of A. Then the columns of A span \mathbb{R}^{m}
Homework EquationsThe Attempt at a Solution
I said that...
1. The problem: A plane mirror of length 2m is kept along the line y=-x as shown in the figure. An insect having velocity of 4 cm/s is moving along x-axis from far away. The time span for which the insect can see its image will be:A)50 sec B)25 sec C)25√2 sec D)50√2 sec
2.The answer...
Problem:
True or False? If $x$ and $y$ are linearly independent, and if $\{\textbf{x}, \textbf{y}, \textbf{z}\}$ is linearly dependent, then $\textbf{z}$ is in Span $\{\textbf{x},\textbf{y}\}$
Solution:
$\textbf{True}$. If $a\textbf{x} + b\textbf{y} = \textbf{0}$ is true and if $a\textbf{x} +...
Hi,
I am designing a concrete roof slab that span across 12m long and 9m width for my living area. The support columns are placed in each 4 corners and they are 300mm x 300mm size. The concrete slab roof is cantilevered out by 1.8m on 3 sides (not all 4 sides) with the thickness of 150mm.
At the...
I'm just having a small trouble understanding the difference ( occurred while I was doing exercise).
A basis is defined as
1)linearly independent
2)spans the space it is found in.
Here is where I get confused:
To determine whether or not a set spans a vector space, I was taught to find its...
Homework Statement
What exactly does it mean for a surface to span a contour/curve? For example, which surface(s) span the contour/curve defined by the equations x^2 + y^2 = 1 and z = y^2?
Homework Equations
N/A
The Attempt at a Solution
I'm not sure if I'm saying something nonsensical, but...
Homework Statement
How to indicate that a vector b is in the span of the columns of a matrix C?
Homework Equations
I could type the definition of Span here, but Wikipedia has it too and it is not necessary or useful now.
The Attempt at a Solution
\mathbf{b} \in \mathrm{Span}\{\mathbf{c}_1...
Hello everyone.
I have to describe is the Severn Bridge, a suspension bridge spanning the River Severn and River Wye in England.
I have found almost all the measurements, except for one: the sag span. I could only find the sag ratio (1/12) but I need to convert this ratio to meters. Can anyone...
This question asks me to prove a statement regarding linear spans. I have devised a proof but I am not sure if it is substantial enough to prove the statement.
1. Homework Statement
Let L(S) be the subspace spanned by a subset S of a linear space V. Prove that if S is a subset of T and T is a...