Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

learn more… | top users | synonyms

0
votes
1answer
7 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 this ...
0
votes
0answers
23 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 ...
0
votes
2answers
36 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 ...
0
votes
2answers
15 views

Inner Product spaces with functions?

I understand inner product space with vectors, but the conversion to functions is throwing me off. Also why do they use an integral here, I've always seen summations. I think I'm missing something ...
0
votes
0answers
22 views

Tensor algebra becomes a graded $R$-algebra short proof verification

I had a post proving that the tensor algebra becomes a graded ring, i have come up with a simple approach that goes as follows: Proposition: The tensor algebra $T(M)$ with multiplication defined ...
0
votes
1answer
14 views

Inner product space related to pythagorean theorem.

So I understand that the an inner product space basically uses pythagorean theorem because it is similar to a distance formula. I'm still having trouble with this proof. I am a bit confused about ...
1
vote
1answer
308 views

Showing no non-trivial t-invariant subspace has a t-invariant complement.

The question is from Hoffman and Kunze Let $T$ be a linear operator on a finite-dimensional vector space $V$. Suppose that: (a) the minimal polynomial for $T$ is a power of an irreducible ...
0
votes
1answer
39 views

Finding length of side on parallelogram

A parallelogram has sides $AB$, $BC$, $CD$, and $DA$. Given $A(1,-1,2)$, $C(2,1,0)$, and the midpoint $M(2,0,-3)$ of $AB$. Find $BD$. I am unsure how to solve this question with the given midpoint ...
2
votes
2answers
35 views

Prove that $T$ is one to one?

I'm very confused on how to do this one. I don't really understand how to find the length of a transformation. Also I'm confused on how that relates to being one to one. Any help appreciated.
0
votes
2answers
105 views

How to differentiate this matrix expression?

I encounter one equation, and want to know how to do the matrix differentiation: ...
0
votes
0answers
10 views

can any one tighten $|e_i^TXe_j|$?

Suppose we have the symmetric matrix $X\in R^{m\times m}$ with its 2-norm $\|X\|_2\leq m$. Then I can get that, for each entry of $X$, $|X_{ij}|=|e_i^TXe_j|\leq\|e_i\|_2\|X\|_2\|\|e_j\|_2=m$, where ...
3
votes
1answer
15 views

A question about biorthogonal basis composed of eigenvectors of a finite-dimensional non-self-adjoint matrix

The non-self-adjoint matrix M has non-degenerate eigenvalues, that is $M \psi_i = e_i \psi_i$, and its adjoint matrix satisfies $M^\dagger \chi_j= e_j^* \chi_j$. I know that $(\chi_j, \psi_i) = ...
1
vote
3answers
52 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 ...
1
vote
2answers
16 views

$R(T)$ and $N(T)$ are $T$-invariant subspaces

Let $T:V\to V$ be linear. Show that $R(T)$ and $N(T)$ are $T$-invariant. I know that $\dim(V)=\dim(N(T))+\dim(R(T))$, but I'm confused on where to go from here.
0
votes
3answers
20 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 ...
0
votes
1answer
71 views

Computing the rotation angle that takes one given basis of $\Bbb R^2$ to another

I am trying to find an angle to rotate the basis $$\left(\begin{pmatrix}1 \\ 0\end{pmatrix}, \begin{pmatrix}0 \\ 1\end{pmatrix}\right)$$ to the basis $$\left(\begin{pmatrix}-1 \\ 1\end{pmatrix}, ...
0
votes
0answers
20 views

Operations with Big Matrices [on hold]

I was able to bring to the scilab the matrices 43x43 using the read command. It Worked! But now i got this error when I try to do operations with the two matrices. -->M.*MD !--error 9999 ...
2
votes
3answers
326 views

Prove that if $\operatorname{rank}(T) = \operatorname{rank}(T^2)$ then $R(T) \cap N(T) = \{0\}$

Let $V$ be a finite-dimensional vector space and let $T:V\to V$ be linear. Prove that if $\operatorname{rank}(T) = \operatorname{rank}(T^2)$ then $R(T) \cap N(T) = \{0\}$. I don't see this ...
0
votes
0answers
12 views

How to compute $proj_wu$; $u$ vector onto $W$ span

Let $u = (1,-2,1,6)$ in $R^4$, and let $W$ = span${(1,1,-1,0),(1,1,0,0)}$ . Compute $proj_wu$ . My Question: Since this is not an orthogonal basis, should I use the Gram-Schmidt process to convert ...
4
votes
1answer
145 views
+50

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 ...
1
vote
4answers
24 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 ...
0
votes
1answer
29 views

Matrix representation of a transformation

We have a linear transformation $T: M_{2\times 2}(F) \to F$ by $T(A) = tr(A)$. We want to compute the matrix representation $[T]$ from $\alpha$ to $\gamma$ coordinates. $M_{2\times 2}$ has the ...
3
votes
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 ...
1
vote
0answers
21 views

Solution of wave equation in Banach space [on hold]

Is it it true laws of physics mandates allowable solution of wave equation; which leads us to work on Hilbert/space? These are also solution in Banach space as well. Is there a practical/physical ...
-4
votes
0answers
50 views

Positive solutions to $A^T A x \geq 0$

Find a positive solution $x$ to the linear inequality $A^T A x \geq 0$. Progress. One special solution is when $A^TA$ is row diagonally dominant, then the column vector $1$ satisfies $A^T A x ...
0
votes
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 :)
0
votes
1answer
16 views

Solving a system of equations containing complex numbers - Gaussian elimination

Problem: Determine the solutions in $\mathbb{C}^3$ of the following system over $\mathbb{C}$: \begin{align*} \begin{cases} 2x+iy-(1+i)z &=1 \\ x-2y+ iz &= 0 \\ -ix +y -(2-i)z &= 1 ...
2
votes
2answers
19 views
0
votes
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. ...
0
votes
1answer
26 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 ...
1
vote
0answers
14 views

Inverse properties of $L_1$ normed matrices

Let's take the adjacence matrix $A$ of a directed graph $G$. We call $\bar{A}$ the row $L_1$ normalized matrix obtained from $A$. (i.e. we divide each elements of the row by the sum of the elements of ...
0
votes
2answers
35 views

Trouble finding Jordan Normal form for $4 \times $ 4 matrix

$M = \left(\begin{array}{cccc}0 & 1 & 0 & 0 \\-3 & 4 & 0 & 0 \\2 & -1 & 2 & 0 \\-1 & 1 & 1 & 2\end{array}\right)$. I find the eigenvalues to be ...
2
votes
3answers
347 views

Distance between two lines by orthogonal projection

I've got the lines' points and vectors $p,q$. My idea was to find a subspace (plane) with the basis of $p,q$ - perpendicular to the lines' axis. Then find the intersecting point $P$ of the lines' ...
2
votes
2answers
28 views

Find the slope of the line that goes through the given points

I know the formula for this type of problem is the second y coordinate subtracted from the first y coordinate over the second x coordinate subtracted from the first x coordinate but for the numbers ...
1
vote
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 ...
2
votes
1answer
32 views

Does a simplex with equal altitudes have to be equilateral?

Consider a simplex in $\mathbb{R}^d$. Assume that all its altitudes have the same length. Does it necessarily mean that the simplex is equilateral, i. e. all distances between its vertices are equal ...
6
votes
2answers
182 views
+100

Linear Algebra: System of Equations

Consider a finite sequence $x_i \in (0,1)$ for $i=1,\ldots, n$ and define $y_i=\dfrac{\Pi_{j=1}^n x_j }{x_i}$. I solved this system for $x$ in terms of $y$ and got $$x_i=\dfrac{\left(\Pi_{j=1}^n y_j ...
3
votes
3answers
598 views

Column Space and SVD

I was reading Gilbert Strang's book and he says that if $A=USV'$ be the SVD of A ( assume square for the moment) then the nullspace of A is given by the last $n-r$ columns of V and the column space by ...
1
vote
1answer
22 views

Writing a matrix in terms of a basis

I've looked for examples but found none similar to this; I have $\mathfrak{sl}(2,K)$ with the given basis $S$ as follows: $S=\{e,h,f\}$ where $e = \pmatrix{0 & 1 \\ 0 & 0}$ $h = \pmatrix{1 ...
4
votes
2answers
85 views

What is the relationship between vector and its associated skew symmetric matrix?

This is my first post in this forum, so hello everyone! I am working with geometries (i.e. areas, volumes and inertias of polygons and polyhedrons in 3D space). For doing that, I to use both the ...
2
votes
0answers
20 views

Did I correctly derive the scheme for this PDE using the Crank Nicolson Method?

I'm taking an Applied Numerical Methods course this semester, and I was given the following homework problem: Basically, before I begin writing any sort of code, I would like to ensure that I have ...
2
votes
1answer
36 views

Least common multiple for integer matrices

Given two full-rank $3\times3$ integer matrices $M_1$ and $M_2$, I am trying to find integer matrices $N_1$ and $N_2$ such that $M_1N_1$=$M_2N_2$, such that $\left|\det(M_1N_1)\right|$ is minimal. ...
1
vote
2answers
25 views

Eigenvalues and eigenvectors for orthogonal projection

I've been self-teaching myself linear algebra using Treil's Linear Algebra Done Wrong and I'm currently stumped on a problem and not sure how to start it. Here is the problem: If someone could give ...
1
vote
1answer
19 views

Linear Algebra Orthogonality Help

I am struggling with this one exercise from self-learning. I simply do not understand what it is asking. If someone could walk me through this problem I would be very grateful.
1
vote
0answers
20 views

Expanding linear functional to base of $V^*$

Given a linear functional $f_1\in V^*$ where $V^*$ is a dual space of $V$, I can expand it to the base of $V^*$ : $B^*=\{f_1,f_2,...,f_n\}$, that I know. But does it mean that exist a base $B$ for $V$ ...
-4
votes
1answer
28 views

Lift-club rates (This should be really easy)

Right, this is actually a real-life problem. I want to join Bob and Joe's lift club. Joe usually pays about \$40 a week (in total) to drive between A and B (for fuel). (Driving from A to B and back is ...
1
vote
2answers
22 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 $T$ ...
0
votes
0answers
33 views

Given $x\in\mathbb R^n$ and an $m\times n$ matrix $A$, show that $x\in \ker A$ or not. [on hold]

Given $x\in\mathbb R^n$ and an $m\times n$ matrix $A$, show that $x\in \ker A$ or not. I understand that the solution to $\ker A$ is the set of all solutions to $Ax=0$. I'm confused about how I ...
1
vote
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 $V^\perp$, ...
1
vote
1answer
32 views

Hyperplanes divide space

Problem. What is maximal number of connected components on which $n$ hyperplanes divides $\mathbb{R}^m$ if they all have 1 common point. In fact this problem was firstly stated in $\mathbb{R}^3$ and ...