What is Linear algebra: Definition and 999 Discussions
Linear algebra is the branch of mathematics concerning linear equations such as:
a
1
x
1
+
⋯
+
a
n
x
n
=
b
,
{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b,}
linear maps such as:
(
x
1
,
…
,
x
n
)
↦
a
1
x
1
+
⋯
+
a
n
x
n
,
{\displaystyle (x_{1},\ldots ,x_{n})\mapsto a_{1}x_{1}+\cdots +a_{n}x_{n},}
and their representations in vector spaces and through matrices.Linear algebra is central to almost all areas of mathematics. For instance, linear algebra is fundamental in modern presentations of geometry, including for defining basic objects such as lines, planes and rotations. Also, functional analysis, a branch of mathematical analysis, may be viewed as the application of linear algebra to spaces of functions.
Linear algebra is also used in most sciences and fields of engineering, because it allows modeling many natural phenomena, and computing efficiently with such models. For nonlinear systems, which cannot be modeled with linear algebra, it is often used for dealing with first-order approximations, using the fact that the differential of a multivariate function at a point is the linear map that best approximates the function near that point.
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⇒Homework Statement
[/B]
Calculate ##S + T## and determine if the sum is direct for the following subspaces of ##\mathbf R^3##
a) ## S = \{(x,y,z) \in \mathbf R^3 : x =z\}##
## T = \{(x,y,z) \in R^3 : z = 0\}##
b) ## S = \{(x,y,z) \in \mathbf R^3 : x = y\}##
## T = \{(x,y,z) \in \mathbf R^3 ...
Homework Statement
Let ##V## be the vector space of the sequences which take real values. Prove whether or not the following subsets ##W \in V## are subspaces of ##(V, +, \cdot)##
a) ## W = \{(a_n) \in V : \sum_{n=1}^\infty |a_n| < \infty\} ##
b) ## W = \{(a_n) \in V : \lim_{n\to \infty} a_n...
Homework Statement
Let V = RR be the vector space of the pointwise functions from R to R. Determine whether or not the following subsets W contained in V are subspaces of V.
Homework Equations
W = {f ∈ V : f(1) = 1}
W = {f ∈ V: f(1) = 0}
W = {f ∈ V : ∃f ''(0)}
W = {f ∈ V: ∃f ''(x) ∀x ∈ R}
The...
I wanted to go through Calculus and then Linear Algebra following either of two paths:
a) Keisler's Infinitesmal approach>>>Nitecki Deconstructing Calculus>>>Nitecki Calculus in 3D>>>Freidberg's Linear Algebra
OR
b) Simmons Calculus with analytic geometry>>>Apostol Vol 1>>>>Apostol Vol...
Dear Fellows,
I have recently completed the study of Stewart's calculus. Next, I want to read Linear Algebra.
I have bought Sheldon Axler's "Linear Algebra done right" textbook. I want to know if my knowledge of calculus is enough to tackle this book or should I first...
Homework Statement
Find the distance from point P (1,7,3) to the line
(x,y,z) = (-2,1,4) + s(1,-3,4),
s is a free variable
Homework Equations
projnQP = ( QP⋅n/(lengthQP)(lengthn) )(n)
The Attempt at a Solution
I'm not quite sure about how to find the normal (n) here, but if I make s=0, I'm...
Homework Statement
From Linear Algebra and Its Applications, 5th Edition, David Lay
Chapter 4, Section 1, Question 32
Let H and K be subspaces of a vector space V. The intersection of H and K is the set of v in V that belong to both H and K. Show that H ∩ K is a subspace of V. (See figure.)...
Homework Statement
If I have two eigenfunctions of some operator, that are linearly indepdendent e.g ##F(x) , G(x)+16F(x) ## and ##F(x)## has eigenvalue ##0##, does this mean that ## G(x) ## must itself be an eigenfunction?
I thought for sure yes, but the way I particular question I just...
Homework Statement
Suppose that we have a system consisting of two interconnected tanks, each containing a brine solution. Tank A contains
x(t) pounds of salt in 200 gallons of brine, and tank B contains y(t) pounds of salt in 300 gallons of brine. The mixture in each tank is kept uniform by...
Homework Statement
Let f1,f2, ..., fn : K -> L be field morphisms. We know that fi != fj when i != j, for any i and j = {1,...,n}. Prove that f1,f2, ..., fn are linear independent / K.
Homework Equations
f1, ..., fn are field morphisms => Ker (fi) = 0 (injective)
The Attempt at a Solution
I...
Homework Statement
Problem- Determine if the set of all function y(t) which have period 2pi forms a vector space under operations of function addition and multiplication of a function by a constant.
What I know- So I know this involves sin, cos, sec, and csc. Also I know that a vector space...
Hi physicsforums,
I am an undergrad currently taking an upper-division course in Quantum Mechanics and we have begun studying L^2 space, state vectors, bra-ket notation, and operators, etc.
I have a few questions about the relationship between L^2, the space of square-integrable complex-valued...
So, I recently came across this example: let us "define" a function as ƒ(x)=-x3-2x -3. If given a matrix A, compute ƒ(A). The soution proceedes in finding -A3-2A-3I where I is the multiplicative identity matrix.
Now , I understand that you can't add a scalar and a matrix, so the way I see it is...
Homework Statement
Suppose that A is a 3 x n matrix whose columns span R3. Explain how to construct an n x 3 matrix D such that AD = I3.
"Theorem 4"
For a matrix A of size m x n, the following statements are equivalent, that is either all true or all false:
a. For each b in Rm, Ax = b has a...
Given an ##N## dimensional binary vector ##\mathbf{v}## whose conversion to decimal is equal to ##j##, is there a way to linearly map the vector ##\mathbf{v}## to an ##{2^N}## dimensional binary vector ##\mathbf{e}## whose ##(j+1)##-th element is equal to ##1## (assuming the index starts...
The question comes out of a corollary of this theorem:
Let B be a symmetric bilinear form on a vector space, V, over a field \mathbb{F}= \mathbb{R} or \mathbb{F}= \mathbb{C}. Then there exists a basis v_{1},\dots, v_{n} such that B(v_{i},v_{j}) = 0 for i\neq j and such that for all...
Homework Statement
Find the Fourier series of the function ##f## given by ##f(x) = 1##, ##|x| \geq \frac{\pi}{2}## and ##f(x) = 0##, ##|x| \leq \frac{\pi}{2}## over the interval ##[-\pi, \pi]##.
Homework Equations
From my lecture notes, the Fourier series is
##f(t) = \frac{a_0}{2}*1 +...
<Moderator's note: Moved from a technical forum and thus no template.>
Hello I am given the following problem to solve.
Identify the quadratic form given by ##-5x^2 + y^2 - z^2 + 4xy + 6xz = 5##.
Finally, plot it.
I cannot seem to understand what I have to do. The textbook chapter on...
Homework Statement
How to prove ##max\{0, \rho(\sigma)+\rho(\tau)-m\}\leq \rho(\tau\sigma)\leq min\{\rho(\tau), \rho(\sigma)\}##?
Homework Equations
Let ##\sigma:U\rightarrow V## and ##\tau:V\rightarrow W## such that ##dimU=n##, ##dimV=m##. Define ##v(\tau)## to be the nullity of ##\tau##...
Homework Statement
Homework Equations
I am not sure. I have not seen the triangle inequality for inner products, nor the Cauchy-Schwarz Inequality for the inner product. The only thing that my lecture notes and textbook show is the axioms for general inner products, the definition of norm...
A problem that I have to solve for my Linear Algebra course is the following
We are supposed to use Mathematica.
What I have done is that I first checked that A is symmetric, i.e. that ##A = A^T##. Which is obvious.
Next I computed the eigenvalues for A. The characteristic polynomial is...
I am trying to understand the geometric intuition of the above equation. ##\rho(\tau)## represents the rank of the linear transformation ##\tau## and likewise for ##\rho(\tau\sigma)##. ##Im(\sigma)## means the image of the linear transformation ##\sigma## and lastly, ##K(\tau)## is the kernel of...
Homework Statement
Let ##W_1=\langle (1,2,3,6),(4,-1,3,6)(5,1,6,12))\rangle## and ##W_2=\langle (1,-1,1,1),(2,-1,4,5)\rangle## be subspaces of ##\Bbb{R}^4##. Find the bases for ##W_1\cap W_2## and ##W_1+W_2##.
Homework Equations...
Homework Statement
Let A be an arbitrary m× n matrix. Find a matrix C such that CA = B for each of the following matrices B.
a. B is the matrix that results from multiplying row i of A by a nonzero number c.
b. B is the matrix that results from swapping rows i and j of A.
c. B is the matrix...
Homework Statement
Find a spanning set for ##P_4##. Find a minimal spanning set. Use Theorem 2.7 to show no other spanning set has fewer elements.
Would simply like someone to check my answers as the book I'm using did not provide a solutions manual. Thank you.
Homework Equations
Theorem 2.7...
Homework Statement
[/B]
Suppose the solution set of some system Ax = b , Where A is a 4x3 matrix, is
*Bold characters are vectors*
x_1= 1 + 3t
x_2 = 2 - t
x_3 = 3 + 2t
where t is a parameter and can be any number.
a) How many pivots are in the row echelon form of A?
b) Let u, v, w be the...
I have not really finished studying linear algebra, I have to admit. The furthest I have gotten to is manipulating matrices a little bit (although I have used this in differential equations to calculate a Wronskian to see if two equations are linear independent, but again, a determinant is...
Let L1 be the line passing through the point P1=(−2,−11,9) with direction vector d2=[0,2,−2]T, and let L2 be the line passing through the point P2=(−2,−1,11) with direction vector d2=[−1,0,−1]T Find the shortest distance d between these two lines, and find a point Q1 on L1 and a point Q2 on L2...
Hi all. So to start I'll say I'm just dealing with functions of a real variable.
In my linear algebra courses one thing was drilled into my head: "Algebraic invariants are geometric objects"
So with that in mind, is there any geometric connection between two orthoganal functions on some...