What is Linear algebra: Definition and 999 Discussions
Linear algebra is the branch of mathematics concerning linear equations such as:
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a
n
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b
,
{\displaystyle a_{1}x_{1}+\cdots +a_{n}x_{n}=b,}
linear maps such as:
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1
,
…
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x
n
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↦
a
1
x
1
+
⋯
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a
n
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{\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.
Hi everybody, I'm a physics student and this is my first post here and i will begin asking one favor:
Anyone could recommend a simple and accessible book about vectors in linear algebra??
I'm already read several books and didn't understand anything.. :cry:
I really need to recover my...
Homework Statement
2: Some proofs:
a) If ##\{ v_1 , v_2...v_n \} ## are linearly independent in a real vector space, so are any subset of them.
b) If any subset of vectors ##\{ v_1 , v_2...v_n \} ## in a real vector space are linearly dependent, then the whole set of vectors are linearly...
Given vector spaces V, W, and a function T:V→W , state the two equations that the function T must satisfy to be a linear function.
Does T:V→W mean a function that maps vectors in V into W? Or what does this actually mean?
I will start calculus 2 in the fall. I am comfortable with the integration techniques. I did not quite get Taylor series an d the like during self study. I can learn this from the professor swing that I have somewhat completed the cal 2 portion and the semester is 2 months away.
now my...
This semester I'm a bit stuck with classes to progress my Electrical Engineering major (having going into it so late), so the only class I can take to progress is a physics course about electricity and the likes. I need at least a three unit class in order to get at least half time so I won't...
Homework Statement
Two glasses.
First glass has 1 L of water.
Second glass has 1 L of alcohol.
Step (1) Pour 1/2 of liquids from glass 1 to glass 2.
Step (2) Pour 1/2 of liquids from glass 2 to glass 1.
What is the limiting situation after 1 billion steps.Homework Equations
N/A
The Attempt...
I am almost finished with the LA chapter in Mary Boas' "Mathematical Methods in the Physical Sciences",. I love the book so far except for the sections in chapter three that she added in the 2005 edition. They are really difficult for me to unpack all by myself. I need a better reference...
Homework Statement .
Let ##A \in \mathbb C^{m\times n}##. Prove that tr##(A^*A)=0## if and only if ##A^*A=0## (here ##0## obviously means the zero matrix).
The attempt at a solution.
By definition of the trace of a matrix, the implication ← is obvious. I am having problems proving...
This is NOT homework.
I am trying to solve a generalized circuit ( each node is a 4-component generalization) and matrices are 4x4 tensors.
And I am trying to write them down -- making common nodes "zero" current, ending up with a matrix as in what's shown in the link...
Homework Statement
Prove the following theorem:
Suppose that B, C, and D are ordered bases for a nontrivial finite dimensional vector space V. let P be the transition matrix from B to C, and let Q be the transition matrix from C to D. Then QP is the transition matrix from B to D...
My friend, who is a beginner in college mathematics, recently asked me to teach her linear algebra.
She has a good grip on High School math.
I am looking for an amusing theorem in linear algebra which can be appreciated by a beginner in college mathematics and at the same time arouse interest...
Homework Statement
Let W be the intersection of the two planes
x + y + z = 0 and x - y + z = 0
In R3. Find an equation for Wτ
Homework Equations
The Attempt at a Solution
So, W = {(x, y, z) l 2y =0}
I don't think that is a correct was to represent W being...
Homework Statement
Let ##V=\mathbb{C}_2## with the standard inner product. Let T be the linear transformation deined by ##T<1,0>=<1,-2>##, ##T<0,1>=<i,-1>##. Find ##T^*<x_1,x_2>##.
Homework Equations
The Attempt at a Solution
Find the matrix of T and then take the conjugate...
Could someone help me with this question? Because I'm stuck and have no idea how to solve it & it's due tomorrow :(
Let S be the following subset of the vector space P_3 of all real polynomials p of degree at most 3:
S={p∈ P_3 p(1)=0, p' (1)=0}
where p' is the derivative of p...
I am currently enrolled in Calculus 1. Will be taking Calculus 2 in summer and Calculus 3 in Fall. I have already registered for these courses. One of the Math electives I have a choice in Fall Semester is either Discrete Math or Linear Algebra. Any suggestions would be greatly appreciated...
Homework Statement
A is an anti-Hermitian matrix.
Show, by diagonalising iA, that:
|det(1+A)|^2 >=1
Homework Equations
A^H denotes hermitian conjugate of A; A^H = -A\
x = Ox' transforms vector components between 2 basis sets.
The Attempt at a Solution
I know that iA is a...
Homework Statement
If M is a real anti-symmetric n x n matrix, M^2 is a real symmetric matrix. Show that M^2 is a non-positive matrix, i.e. x(transposed) M^2 x <= 0, for all vectors x.
Homework Equations
det(M) = (-1)^n det (M)
The Attempt at a Solution
I attempted to use the...
Just came across LU decomposition and I am not sure how to work on this problem:
Let L and L1 be invertible lower triangular matrices, and let U and U1 be invertible upper triangular matrices. Show that LU=L1U1 if and only if there exists an invertible diagonal matrix D such that L1=LD and...
Homework Statement
Find equation of plan H in R^4 that contains the point P= (2,-1,10,6)
and is parallel to plain H2: 4a +4b + 5c-6d = 3 then answer the following questions:
A. find normalized normal of plane H which has an angle theta with the normal n= (4,4,5,-6) of H2 such that...
Homework Statement
We have seen that the linear transformation ##T(x_1,x_2)=(x_1,0)## on ##\mathcal{R}^2## has the matrix ##A = \left( \begin{smallmatrix} 1&0\\ 0&0 \end{smallmatrix} \right)## with respect to the standard basis. This operator satisfies ##T^2=T##. Prove that if...
Consider the following three vectors in R3: u1=(3,6,2) , u2=(-1,0,1) , u3=(3,λ,7)
a) Find all values of λ E R, such that {u1, u2, u3} spans R3, i.e.R3 = span {u1, u2, u3}
b) Find the value of λ E R, such that {u1, u2, u3} spans a plane in R3.
c) Find all values of k E R, such that the...
Homework Statement
Let A be the matrix
A =
1 −3 −1 2
0 1 −4 1
1 −4 5 1
2 −5 −6 5
(a) Find basis of the column space. Find the coordinates of the dependent columns relative
to this basis.
(b) What is the rank of A?
(c) Use the calculations in part (a) to...
Hi,
I'm reading Shilov's linear algebra and in part 2.44 he talks about linear independent vectors in a subspace L which is a subset of space K( he refers to it as K over L). I don't understand why he says that a linear combination of vectors of the subspace L and vectors of the subspace K...
Homework Statement
If dim(X)=n, show that the vector space of k-linear forms on X is of dimension nk.Homework Equations
The Attempt at a Solution
So I know we need to let x1, x2,...xn be a basis for X. My professor then said to "show that the function fj1,...,jk, 1≤jl≤n defined by...
Homework Statement
Let T: C∞(R)→C∞(R) be given by T(f) = f'''' where T sends a function to the fourth derivative.
a) Find a basis for the 0-eigenspace.
b) Find a basis for the 1-eigenspace.
The Attempt at a Solution
I just want to verify my thought process for this problem. For...
The time has come to schedule for next semester's classes. I will be a senior in physics and choosing some electives. I am trying to decide between taking matrix theory (linear algebra) or graduate level classical mechanics. I really WANT to take the mechanics course but I feel that maybe I...
Hi everyone.
Long time lurker first time poster.
I am taking a linear algebra class this semester and I am really struggling.
Does anyone have any good resources for learning/understanding function spaces, polynomial interpolation and orthogonal expansion etc..?
Thanks
this probably is a wrong section but i did not know where else to put it (feel free to move it to where it belongs).
that said could someone recommend me a pure math linear algebra book? preferably with LOTS and LOTS of examples! i think i have decent grasp of the basic theory but i have been...
We use the term VECTOR in vector algebra, in vector calculus, in Linear Programming and in Linear algebra. For vector algebra and vector calculus, by a VECTOR we mean a quantity having both magnitude and direction. Does it have same meaning in the context of Linear Programming and in Linear...
Are there any Introductory Linear Algebra books out there that are not so proof-laden? All those proofs only make things more complicated and I would rather just learn applications of Linear Algebra rather than sit through a bunch of long proofs.
Can anyone suggest some books to me for this...
Homework Statement
if V = C_x for some x belongs to V then show
deg(u_L) = dim(V)
here,
L: V -> V linear operator on finite dimensional vector space
C_x = span {x, L(x), L^2(x),....}
u_L = minimal polynomial
Homework Equations
The Attempt at a Solution
since...
Homework Statement
This is a question from an upper level econ course that is giving me quite a bit of trouble. Fluency in linear algebra is assumed for the course. I'm taking a linear algebra course for the first time this semester so I'm still scrambling to learn the basics. If anyone has a...
Homework Statement
Let T: C→D with dim(A)=n and dim(B)=m. Show that there exists bases B and B' for C and D, respectively, such that the matrix of T in block form is
M=|I 0|
|0 0|
where I is a k by k identity matrix
Homework Equations
The Attempt at a Solution
Honestly no idea...
Homework Statement
find a formula for \begin{bmatrix}
1 & 1& 1\\
0& 1& 1\\
0& 0 & 1
\end{bmatrix} ^n
and prove it by induction
the induction part is ok.
I'm just having trouble finding a pattern
I may have figured it out but it looks too cumbersome
Homework Equations...
Homework Statement
A. If the equation Ax=0 has only the trivial solution, then A is row equivalent to the nxn identity matrix.
B. If the columns of A span R^n, the columns are linearly independent.
C. If A is an nxn matrix, then the equation Ax=b has at least one solution for each b in R^n...
Homework Statement
Let S : U →V and T : V →W be linear maps.
Given that dim(U) = 2, dim(V ) = 1, and dim(W) = 2, could T composed of S be an isomorphism?
Homework Equations
If Dim(v) > dim(W), then T is 1-1
If Dimv < dim(w), then T is not onto.
The Attempt at a Solution
So...
How difficult will this be? I have 6 semesters left at CC and I need to take college algebra, precalc, calc 1, calc 2, calc 3, linear algebra, and dif equations.
So I have to have some overlap somewhere but I am not 100% sure. College Algebra is a prereq for precalc, which I have to take...
I'm doing my homework but I'm lost on one thing. Let's say that we have a systems of equations like so:
2x1+3x2=y1
4x1+2x2=y2
Instead of setting it to a constant our teacher sets it to a variable, he says that to be able to compute this, the augmented matrix should look like:
2 3|1 0
4 1|0 1...
Homework Statement
M: V -> V linear operator st M^2 + 1_v = 0
find the POSSIBILITIES for min. pol. of M^3+2M^2+M+3I_v
Homework Equations
The Attempt at a Solution
using M^2 = -1_v,
i rewrote the operator(?) as
M^3 + M + I_v
i don't know what to do. i guessed min poly to...
Homework Statement
Show that B|A|+A|B| and A|B|-B|A| are orthogonal.
Homework Equations
N/A
The Attempt at a Solution
I'm not too sure exactly how to start this. I do know that for two things to be orthogonal, the dot product has to be equal to 0, but I'm not sure how to evaluate...
Homework Statement
If llull = 4, llvll = 5 and u dot v = 10, find llu+vll. u and v are vectors
Homework Equations
llu+vll = llull + llvll cauchy schwarz
The Attempt at a Solution
(1) llu+vll = llull + llvll
(2) (llu+vll)^2 = (llull + llvll)^2
(3) (llu+vll)^2 = llull^2 +...
Homework Statement
Here is a picture of the diagram.
http://gyazo.com/f1b7051fda5b9e1d3a185c53abde1211
I must use the Gauss Jordan elimination method and solve for X1, X2 and X3
I am having problems setting up my equations
Homework Equations
The Attempt at a Solution...
Homework Statement
Consider the system:
(1) 6x + ky = 0
(2) 4x + 6y = 0
The system will have a unique solution when k is:
(a) equal to 9
(b) any real number
Which statements are true.
Homework Equations
(1) 6x + ky = 0
(2) 4x + 6y = 0The Attempt at a Solution
If m=n (number of equations is...
Some background: I am self studying dynamics and I have encountered a fundamental problem with either my understanding of linear algebra, or I am just plain dumb. So, I print screened the page of the book we're on. Now let me try to reduce some ambiguity in my question, I have a general...
forgive the messiness; i take bad notes in class.
http://i.imgur.com/VmW8Ubg.jpg
towards the middle of the page where it says "this is equivalent to..." and then my professor wrote what follows but i thought the row vector should be complex conjugates? ie, the red writings are not actually...