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 (1)

1
vote
2answers
31 views

How to prove this result?

Let {$\Delta_1,\Delta_2,\Delta_3\cdots\cdots\cdots\cdots\Delta_n$} be the set of all determinants of order 3 that can be made with the distinct real numbers from set $S=\{1,2,3,4,5,6,7,8,9\}$. Then ...
1
vote
4answers
47 views

Proving an Identity with Scalar Product

How to prove following: $$(u,v) = \frac{1}{2} \left\{ (u+v,u+v) - (u,u) - (v,v) \right\}$$ Hint: This identity implies that the following 4 properties define scalar product uniquely: $i. (u,v) = ...
1
vote
2answers
42 views

In $\mathbb{R}^2$, u orthogonal to v, v orthogonal to w => u parallel to w

Question: Given $u,v,w\in\mathbb{R^2}$ non-zero vectors. If $u\perp v$ amd $v\perp w$, then $u\parallel w$. I think this question might be stupid but I couldn't prove it using only algebra. Can ...
1
vote
2answers
76 views

Solving augmented linear system $\left(\begin{smallmatrix}1&0&-2&~~~&0\\0&1&0&&0\\0&0&0&&0\end{smallmatrix}\right)$

$$\left(\begin{smallmatrix}1&0&-2&~~~&0\\0&1&0&&0\\0&0&0&&0\end{smallmatrix}\right)\to ...
1
vote
4answers
170 views

A Proof that Orthogonal Complement is unique

So our professor asked us to prove that considering any subspace $S$ of a vector space $V$, the orthogonal complement $S^{\perp}$ is unique. I have devised a proof and I am not sure whether this ...
1
vote
1answer
104 views

Looking for fast computation method of $Ax=b$ ($A$ is sparse matrix)

I am looking for fast method to solve linear equation $$Ax=b$$ In which A is sparse matrix. Could you suggest to me some current method for this task. Thank in advance
1
vote
2answers
36 views

$a_1\textbf{v}_{1} + a_2\textbf{v}_{2} = b_1\textbf{v}_{1} + b_2\textbf{v}_{2} $ if and only if $a_1 = b_1$ and $a_2 = b_2$?

Would it be correct to say that $a_1\textbf{v}_{1} + a_2\textbf{v}_{2} $ is equal to $b_1\textbf{v}_{1} + b_2\textbf{v}_{2} $ if and only if $a_1 = b_1$ and $a_2 = b_2$? I feel as though it is a very ...
1
vote
2answers
366 views

How to define an affine transformation using 2 triangles?

I have $2$ triangles ($6$ dots) on a $2D$ plane. The points of the triangles are: a, b, c and x, y, z I would like to find a ...
1
vote
2answers
101 views

Relation between volume form and cross product

Euclidean three-dimensional space (it's simpler). Defining $\eta={e^*}^1 \wedge {e^*}^2 \wedge {e^*}^3$, with $\{{e^*}^1,{e^*}^2,{e^*}^3\}$ dual of the orthonormal basis, and indicating the classic ...
1
vote
2answers
66 views

Direct sum of modules, is there something like direct difference?

This: $U_1 \bigoplus U_2 = V \Leftrightarrow U_1 = V - U_2$ is probably not true, but is there an alternative for writing this?
1
vote
3answers
97 views

To determine Nullity of $T$

Let $V$ be vector space of polynomials of degree $\leq n$ . And $ T : V \rightarrow \mathbb R ^{m}$ be defined as $T (P (x)) = (P (1) , P (2) ,..., P (m) )$ I have to determine nullity of $T$ . ...
1
vote
2answers
44 views

Showing $(Tp)(x) = x^2p(x)$ is a linear map (transformation)

Define a linear map function $T: \mathcal{P}(\mathbb{R}) \to \mathcal{P}(\mathbb{R})$ where $\mathcal{P}(\mathbb{R})$ is the set of all polynomials with real-valued coefficients. Now let $T$ belong to ...
1
vote
3answers
120 views

Intuitive transition from matrices to tensor-concept

I would like to know how to build intuition for the concept of a tensor using the following reasoning: If I conceive of a vector as an extension of the scalar concept, i.e. an $N \times 1$ "array of ...
1
vote
2answers
70 views

Using inverse of matrix A as approximate inverse of matrix that is very close to A

Say we have two matrices, $A$ and $A'$ so that $A\approx A'$, and we have the inverse of $A$, $B$, where $AB=I$, and the inverse of $A'$ where $A'B'=I$. If we have some guarantee about how big any ...
1
vote
3answers
81 views

Question on the definition of vector spaces.

My question is perhaps useless, but I want to shed some clarity on this matter. I'm bothered by people that say a vector space is a "bunch of vectors". Or that a vector space "consists of ...
1
vote
1answer
133 views

Proof of De Moivre's theorem using generating functions

I've come across the following proof of De Moivre's theorem: $$ \cos(n\theta) + i\sin(n\theta) = ( \cos\theta + i \sin\theta )^n $$ The proof establishes that: $ \forall \ \ |s| <1 $, $$ ...
1
vote
3answers
46 views

Equations with variable powers

Find the roots of the equation $3^{x+2}$+$3^{-x}$=10 . By inspection the roots are $x=0$ and $x=-2$. But how can I solve this equation otherwise?
1
vote
2answers
131 views

What are examples of two non-similar invertible matrices with same minimal and characteristic polynomial and same dimension of each eigenspace?

I'm trying with matrices over $\mathbb F_2$ and trying to have a look at the Jordan canonical forms of these matrices. If the size of the biggest Jordan block is the same with 1's in all diagonal ...
1
vote
2answers
78 views

A symmetric matrix with eigenvalues all $0$ or all $1$: does it equal $0$ or identity?

I have these general wondering about matrices but I don't know to proceed with a proof or a counter example. Suppose that $A$ (dimension $n\times n$) is a real symmetric matrix. If $A$ has $n$ ...
1
vote
2answers
65 views

Determing whether (S, *) is an Abelian group

Given a set defined as $S=\{(a,b) | a,b \in \mathbb{Q} \land a^2+b^2=1 \}$ and a binary operation $*$ defined as $(\forall(a,b),(c,d) \in S) ((a,b)*(c,d) = (ac-bd, bc+ad))$, determine whether $(S,*)$ ...
1
vote
3answers
369 views

Prove that $ AA^T=0\implies A = 0$

Let $A$ be an $n \times n$ matrix with real entries, where $n\geq2$. Let $AA^T = [b_{ij}] $, where $A^T $ is the transpose of $A$. If $b_{11} + b_{22 }+\cdots+ b_{nn} = 0$, show that $A = 0$. ...
1
vote
2answers
95 views

establish the identity $\|u+v\|^2+\|u-v\|^2=2\|u\|^2+2\|v\|^2$ for $u$, $v \in \mathbb R^n$

establish the identity $\|u+v\|^2+\|u-v\|^2=2\|u\|^2+2\|v\|^2$ for $u$, $v \in \mathbb R^n$. I couldn't understand how to solve it please just give me the first step, maybe I can figure out the ...
1
vote
3answers
77 views

$x-y-2z=0$ find a perpendicular vector

Why is the vector $e=(1,-1,-2)$ ?
1
vote
1answer
64 views

If $A$ is skew-symmetric, then a fixed row/column operation produces a new skew-symmetric matrix

Suppose $A$ is a skew-symmetric matrix. Fix an elementary row operation. If we carry out this row operation on $A$, and then carry out the corresponding column operation on the resulting matrix, do we ...
1
vote
1answer
93 views

Prove that if $p\le n$, then $p$ does not divide $n! + 1$

I'm having trouble on how to approach this problem Prove that if $p\le n$, then $p$ does not divide $n! + 1$ ($p$ is prime and $n$ is an integer).
1
vote
1answer
165 views

If det(A) is zero, what is det(adj(A))?

I wanted to prove that det(adj(A))=det(A)^n-1 for an nxn matrix A. I separate the proof in two cases: singular and non-singular matrix A. For the non-invertible A, det(A)=0. In my head, I know that ...
1
vote
2answers
30 views

Proof with Positive Symmetric Matrices

Prove that if $K_1$ and $K_2$ are positive definite $n × n$ matrices, then $$K = \begin{pmatrix}K_1& 0\\ 0 &K2\end{pmatrix}$$ is a positive definite $2n × 2n$ matrix. Is the converse true? I ...
1
vote
1answer
73 views

Eigenvalues of linear operator TS and ST for infinite dimensional space

Here is the original problem: Let $S$ and $T$ be linear operators on a finite-dimensional vector space $V$. Show that $TS$ and $ST$ have the same eigenvalues. I can prove it. However, my question is: ...
1
vote
2answers
96 views

finding f with the given information, antiderivatives?

Find $f$. $$f''(t)=3/\sqrt{t}$$ $$f(4)=12$$ $$f'(4)=5$$ I'm not quite sure how I am supposed to find $f$ with this information.
1
vote
2answers
84 views

Calculate formula for $n^{th}$ power of a matrix

How would I find a formula for the $n^{th}$ power of the matrix $$\begin{pmatrix} 1&2\\ 3&4 \end{pmatrix}? $$ Is there a way to do this for all matrices?
1
vote
2answers
86 views

Linear transformation is injective if and only if there exists a linear transformation where the composition is identity

How to prove that if $U$ and $V$ are finite dimensional vector spaces and $T:U \rightarrow V$ is a linear transformation, then $T$ is injective if and only if $S \circ T$ is the identity function on ...
1
vote
2answers
43 views

Let $A$ be a $3×4$ matrix. Estimate $\det(A'A)$ and $\det(AA')$

Let $A$ be a $3×4$ matrix. Estimate $\det(A'A)$ and $\det(AA')$. I would first assume that $A$ has rank $3$. Then $A'A$ would be a $4\times 4$ matrix with rank $3$ and therefore it would have ...
1
vote
1answer
47 views

Is the matrix inequality $P > Q \geq 0$ implies $P^2 > Q^2$?

Assume $P$ and $Q$ are positive definite and positive semi-definite, respectively, and both are symmetric. Then, is it true that \begin{equation} P > Q \;\; \Longrightarrow \;\; P^2 > Q^2 ...
1
vote
2answers
64 views

complex eigenvalues and invariant spaces

I am currently reading Guillemin and Pollack's Differential Topology, and the following claim is made without proof: Given a linear isomorphism $E: \mathbb{R}^k \to \mathbb{R}^k$, with $k>2$ and ...
1
vote
2answers
52 views

Matrix Multiplication Commutativity Generalization.

Find condition for $a, b, c$ and $d$ such that the matrix $B= \begin{bmatrix} a & b \\ c & d \\ \end{bmatrix}$ commutes with any two by two matrix. My attempt: Commute means $AB=BA$. I ...
1
vote
1answer
72 views

How to find the determinant of this $(2n+2)$ x $(2n+2)$ matrix?

I need to calculate the determinant of the following matrix:$$\begin{bmatrix}0&0&-2x_1& \cdots &-2x_n&0& \cdots &0\\0&0&0& \cdots&0&-2x_1& ...
1
vote
1answer
88 views

Determine whether the given the orthogonal matrix represents a roation or reflextion…?

I am given the matrix $$ \begin{bmatrix} -1/2 & \sqrt{3}/2 \\ -\sqrt{3}/2 & -1/2\\ \end{bmatrix} $$ I think this is a reflection because I tied sketching a ...
1
vote
3answers
93 views

$U(R) \equiv U(B) \equiv U(N)$, trying to find unique values that result in indifference.

I'm currently working on a decision problem, and for some reason I am struggling with a system of equations, which should be the easiest part of the problem. The correct answers are ...
1
vote
3answers
66 views

Show that $V=ker(P)+im(P)$ and $ker(P)\cap im(P)=\{0_v\}$

If $P:V\rightarrow V$ is a linear map such that $P^2=P$, show that show that $V=\text{ker }(P) + \text{im }(P)$ and $\text{ker}(P)\cap \text{im }(P)=\{0_v\}$. So I know that $\text{ker }(P)+\text{im ...
1
vote
3answers
42 views

How did my book see the rank and what is wrong with my null space

Reduced Row Echleon form: $$\begin{bmatrix} 3 & 1 & 3 & -2 \\ 2 & -1 & 4 & -5 \\ 0 & 0 & 0 & \theta+6 \\ 0 & 0 & 0 & \theta +6 \end{bmatrix}$$ ...
1
vote
4answers
56 views

Prove that for any nonzero vectors $\bf u$ and $\bf a$ in $\Bbb R^n$, the vector $\bf a$ is orthogonal to ${\bf u} - \mathrm{proj}_{\bf a}{\bf u}$.

Prove that for any nonzero vectors $\bf u$ and $\bf a$ in $\Bbb R^n$, the vector $\bf a$ is orthogonal to ${\bf u} - \mathrm{proj}_{\bf a}{\bf u}$. I'm not sure how to start proving this. I don't ...
1
vote
2answers
125 views

Minimal polynomial of the operator $T:V\oplus W\to V\oplus W$

Let $V$ and $W$ be two finite dimensional vector spaces over $\Bbb R$ and let $T_{1}:V\to V$ and $T_{2}:W\to W$ be two linear transformations whose minimal polynomials are given by ...
1
vote
2answers
43 views

Does the line $(2,1,1)+t(-3,1,5)$ live within the plane $31x+3y+18z=62$?

I have a doubt with this exercise: Have the plane $$31x+3y+18z=62$$ What is the distance between this plane and some line $(x,y,z) = (2,1,1) + t(-3,1,5)$ for some $t\in\mathbb{R}$? The ...
1
vote
2answers
130 views

If $A^2 =0$ then possible rank of $A$

Let, $A$ be a non zero matrix of order $8$ with $A^2 =0.$ Then one of the possible value for rank of $A$ is (a) $5$ (b) $4$ (c) $6$ (d) $8$. Attempt : As , $A^2=0$ , so $A$ is a nilpotent ...
1
vote
2answers
89 views

If $A$ is $n\times n$ matrix with $(A-I)^2=0$ then which of the following is true?

If $A$ is $n\times n$ matrix with $(A-I)^2=0$ then which of the following is true? $1.$ $A=I$ $2.$ $\det(A)=1$ $3.$ $\operatorname{trace}(A)=n$ I have counter example for the first option.For ...
1
vote
1answer
54 views

Solving $Ax=B$: what's wrong with this linear algebra argument?

With $K>L$ and assuming that we are working with real variables, suppose that $B$ is $K\times 1$ and $A$ is $K\times L$ with full column rank. I'm trying to find $x$ ($L\times 1$) satisfying: $$ ...
1
vote
1answer
60 views

Gram matrix to be cancelled

Let $V$ be a $n$ dimensional Euclidean space with inner product $<\cdot,\cdot>$, with basis $e_1,\cdots,e_n$. Then the Gram matrix is $A=(a_{ij})$ with $a_{ij}=<e_i,e_j>$. It is well-known ...
1
vote
2answers
37 views

Example of two subspaces $W_1$ and $W_2$ of $ V$ such that $W_1∪W_2$ is also a subspace of $V$

So I know the obvious counter example would be to let: $W_1 = \{(a, 0) | a \in\mathbb{R}\}$ and $W_2 = (0, 0)$. Where $W_1 + W_2 = (a, 0)$ which is an element of $W_1\cup W_2$. But if I wanted ...
1
vote
2answers
41 views

Matrix multiplication of inverse and non-inverse

I just want to ask why $A^{-1} B^{-1} = B^{-1} A^{-1}$ but not $AB = BA$? Aren't the two essentially the same? Why are they different? Also what does $(A + B)^{-1}$ essentially equal to? I know that ...
1
vote
4answers
127 views

Linear Algebra - Prove Isomorphism.

Let $T : \Bbb R^n \rightarrow \Bbb R^n$ Linear transformation. Prove that there is a real number $\alpha$ that the transformation $\alpha I-T$ is isomorphism. isomorphism is only if $\ker T={0}$ or ...