Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them. This includes: systems of linear equations, basis, dimension, subspaces, matrices, determinant, trace, eigenvalues and eigenvectors, diagonalization, Jordan form, etc. For questions specifically ...

learn more… | top users | synonyms

0
votes
2answers
36 views

Characteristic polynomial and eigenvalues of a $3 \times3$ matrix.

Hi so I have to find the characteristic polynomials and the eigenvalues of the matrix: $$A = \begin{bmatrix}1 & 0 & 3\\2 & -2 & 2\\3 & 0 & 1\end{bmatrix}$$ So I know you use ...
0
votes
1answer
15 views

Determining if Linear transformation

Please help me get and understand this concept of linear algebra based on this questions: Determine whether or not $T$ is a linear transformation from $\Re^2$ to $\Re^2$ if u$ \in \Re^2 $ and v$ \in ...
2
votes
2answers
43 views

Rewriting the matrix equation $AX = YB$ as $Y = CX$?

Is it possible in general, if $A,B,C,X,Y$ are square and of the same dimensions? If so, does it generalize to non-square matrices (using a pseudoinverse)? I'm doing some curve fitting in which I have ...
0
votes
1answer
49 views

Characterize matrices A such that trace(AC)=0 for every matrix C with trace(C)=0

$A$ is an $n\times n$ matrix on the field $F$ such that for every $n\times n$ matrix $C$ with $\operatorname{trace}(C)=0$ we have $\operatorname{trace}(AC)=0$. Can we characterize such matrices $A $? ...
0
votes
0answers
11 views

Bilinear form equations

$\mathit{V}$ is vector space over complex, $R_A, R_B, R_C\in M_3(\mathbb{C})$ are conjugate symmetric positive definite matrix. Know $R_A,R_B,R_C$ and find vectors $X_1, X_2, X_3$ such that $$ ...
0
votes
0answers
19 views

Which casses of matrices contain A and which contain B? Linear Algebra

Am pretty confused about classes, I don't know what it means, so so I can't really do part_A and I need your help with it? For part B, I got all eigen = 1 for matrix A, and 0 for matrix B, Is this ...
2
votes
2answers
24 views

The inverse of a linear transformation $A$ can be expressed as a polynomial in $A$

Suppose that $A$ is a non-singular linear transformation of an $n$-dimensional linear space into itself. Show that there exists some polynomial $c_0+c_1z+\ldots+c_kz^k$ so that ...
0
votes
1answer
21 views

Fulton's algebraic curves question 1.36

I'm trying to solve the question 1.36 from Fulton's algebraic curves book: Let $I=(Y^2-X^2,Y^2+X^2)\subset\mathbb C[X,Y]$. Find $V(I)$ and $\dim_{\mathbb C}(\mathbb C[X,Y]/I)$ Obviously ...
2
votes
0answers
23 views

Method of orthogonalization that preserves invertibility

Is there a method of orthogonalization such that, given an invertible matrix $A$ with entries in the real numbers, applying the method and then inverting the result is the same thing as applying the ...
0
votes
0answers
30 views

What is the linear combination of B?

I have a problem where I am finding $A^n$B where B=$[3,1,1]^t$. I know the steps in solving, but I do not remember how to find linear combination. I do not see it. There has to be a way to calculate ...
1
vote
0answers
9 views

Trouble understanding Hoffman / Kunze exercise [duplicate]

I am finishing up a number theory class and will be studying graduate Linear Algebra in the fall so I thought I'd start early getting familiar with the text and authors by doing some of the early text ...
0
votes
0answers
11 views

How do you get nullspace N(A) to be orthogonal to C(A^H)

In the picture below, C(A) is given in number7, but I am doing number_8. Ii did a gauss jordan where by i subtracted R2-iR1 to get 0 belo 1st pivot and 1 as the second pivot in column2, row2. Then I ...
1
vote
1answer
20 views

Particular solution to the matrix form Ax=b

This question is more for general understanding than looking for a specific answer. I have a theorem that states: "If Ax=b has a solution x$_p$, then the general solution to the equation is x$_p$ + ...
3
votes
0answers
25 views

Coproduct in the category of vector spaces with bilinear forms

I'm trying to work out the coproduct in the category of (say real) vector spaces equipped with bilinear forms, where the morphisms $(V,b) \to (V',b')$ are the linear maps $T : V \to V'$ such that $T^* ...
1
vote
1answer
25 views

Prove existence of Diagonalizable Matrix

Suppose R, T $\in L(F^3)$ each have 2, 6, 7 as eigenvalues. Prove that there exists an invertible operator S $\in L(F^3)$ such that $R=S^{-1}TS$. What I got so far is that since R and T have three ...
1
vote
2answers
32 views

Why use Gauss Jordan Elimination instead of Gaussian Elimination, Differences

Why use Gaussian Elimination instead of Gauss Jordan Elimination and vice versa for solving systems of linear equations? What are the differences, benefits of each, etc.? I've just been solving ...
0
votes
3answers
45 views

Orthogonal diagonalization of Symmetic Matrices

Step 1: find the symmetric matrix A which represents q and find its characteristic polynomial \Delta (t). Step 2: find the eigenvalues of A which are the roots of \Delta (t). Step 3: for each ...
1
vote
3answers
65 views

Proof that $\mathrm{ker}(A^{T}A) = \mathrm{ker}(A)$?

Is there a proof that can help me understand why this is the case? I can't conceptualize the reason for this in my mind. Thanks.
0
votes
1answer
40 views

Eigenvectors for normal operators and their adjoints

Can someone tell me if this proof is correct? Claim:V is a vector space over the Complex field. $T:V\rightarrow V$ is a normal operator. Then if $v\in V$ is an eigenvector with the eigenvalue ...
0
votes
1answer
49 views

If there is the inverse operator of the operator A, then $(A^{-1})^{-1}=A$?

A friend of mine asked me today for this example: If there is the inverse operator of the operator A, then $(A^{-1})^{-1}=A$? But I do not have the ability to help, so I told him that his example ...
0
votes
2answers
35 views

Linear Programming with 3 variables

What is the optimal solution for maximizing $X_1 + 2X_2 + 3X_3$ subject to the constraints that: $X_1 + X_2 + X_3 \leq 9$, $-X_1 + 2X_2 + 5X_3 \leq 15$, $X_1 \geq 0$ $X_2 \geq 0$. My answer is ...
1
vote
2answers
53 views

How does differentiation work on matrices?

I am studying on K-Means clustering. I am wondering how the differenciation work on the matrices circled in red below in the image. Can I have your expertise on how the calculations done in the ...
3
votes
6answers
337 views

Are inverse matrices unique?

Does a matrix have only one inverse matrix (like the inverse of an element in a field)? If so, does this mean that $A,B \text{ have the same inverse matrix} \iff A=B$?
1
vote
2answers
18 views

Finding the vertices of a square - straight lines

Question: Each side of a square is of length $6$ units and the center of the square is $(-1, 2)$. One of its diagonals is parallel to $x + y = 0$. Find the co-ordinates of the vertices of the square. ...
3
votes
1answer
57 views

Surprising necessary condition for a “shift-invariant” determinant

Let $A$ be a $4\ x\ 4$ binary matrix and $Z=\pmatrix {s&s&s&s \\ s&s&s&s \\s&s&s&s \\s&s&s&s}$ Then $\det(A+Z)=\det(A)=1\ $ (independent of s, so ...
5
votes
4answers
139 views

If $\lambda$ is an eigenvalue of $A^2$, then either $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of $A$

$A$ is an $n\times n$ matrix of complex numbers. Prove that if $\lambda$ is an eigenvalue of $A^2,$ then $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of $A.$ If $\lambda$ is an eigenvalue ...
2
votes
1answer
40 views

Help with simple rotation on an x,y plane

I'm a programmer, with too little background in mathematics, and I am currently faced with the challenge of rotating an object on a 2 axis plane. Something that is hopefully quite easy for you guys. ...
-3
votes
0answers
55 views

Series expansion - Involving matrix exponent [on hold]

I have a skew symmetric matrix $$C=\left( \begin{array}{ccc} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \\ \end{array} \right).$$ Given Conditions $x = ...
0
votes
1answer
22 views

Matrices as sets of vectors

What exactly does it mean when someone says a matrix may be intrepreted as a set of vectors? As in: "A matrix can be considered a set of vectors, organised as rows or columns" It seems it would only ...
0
votes
1answer
27 views

Diagonalization of Skew symmetric matrix

I have a skew symmetric matrix $$C=\left( \begin{array}{ccc} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \\ \end{array} \right).$$. and we have the relation $C=UDU^{-1} ...
1
vote
3answers
52 views

If $A,B$ are invertible so $AB$ is invertible

I thought about the following proofs but I am not sure about them there is $C,D$ so that $AC=CA=I$ and $BD=DB=I \rightarrow CABD=I \rightarrow$ due to associativity roles is no matrix $E$ so that ...
0
votes
1answer
22 views

Is it necessary for a linear map to be an automorphism to allow polar decomposition?

Bowen and Wang's Introduction to Vectors and Tensors I (pg. 168) states a general form of the polar decomposition theorem as Every automorphism A has two unique multiplicative decompositions $$ ...
0
votes
2answers
61 views

Prove Or Disprove: tr(AB)=tr(A)*tr(B)

$\mathrm{tr}(AB)=\sum\limits_{i=1}^n \sum\limits_{j=1}^n a_{ij}*b_{ji}$ $\mathrm{tr}(A)*\mathrm{tr}(B)=\sum\limits_{i=1}^n a_{ii}*\sum\limits_{i=1}^n b_{ii}$ Therefore $\mathrm{tr}(AB) \neq ...
0
votes
1answer
41 views

LU factorisation question

I have a question from a past exam with the solution but i am getting a completely different answer to that of the solution. Could someone please tell me where I am going wrong? question: Find the ...
0
votes
4answers
59 views

Give a Counterexample if V is infinite dimensional

$V = nullT \oplus rangeT$ if and only if $V = null T + rangeT$. Where $T \in L(V)$ I'm having alot of trouble coming up with an example for this. Shouldn't both cases always fail if V is infinite ...
2
votes
1answer
52 views

Can polynomials with degree at least 2 over $\mathbb{R}$ have finite number of solutions in $End(\mathbb{R},+)$

Consider a polynomial of degree at least 2 with all coefficients in $\mathbb{R}$. We are concern with set of solution for the polynomial in $End(\mathbb{R},+)$ - the endomorphism ring of the abelian ...
0
votes
3answers
30 views

Prove that T = I with Linear Transformations.

Suppose that $T \in L(V)$ and $T^2 = I$ and -1 is not an eigenvalue of T. Prove that T = I. What I tried was: Suppose $\lambda$ is an eigenvalue of T such that $T(v) = \lambda v$ Then we know that ...
1
vote
2answers
26 views

Applying a polynomial to an operator?

Suppose $T \in L(V)$ and $\exists$ a positive integer n such that $T^n = 0$. Prove that $(I-T)$ is invertible and that $(I-T)^{-1} = I + T + \dots + T^{n-1}$. I wish I could say that I attempted ...
1
vote
2answers
27 views

Prove that these have the same eigenvalues

Suppose $T \in L(V)$. Suppose $S \in L(V)$ is invertible. Prove that $T$ and $S^{-1}TS$ have the same eigenvalues. What is the relationship between the eigenvectors of T and the eigenvectos of ...
0
votes
1answer
30 views

Why do these vectors not span the given space?

I need some help understanding this solution to a problem. I am working on the problem above. I know that in order for a set of vectors to be a basis it must be linearly independent and span the ...
0
votes
2answers
22 views

Finding the possible dimensions of the intersection of subspaces

I read this problem, If $V$ and $W$ are 2-dimensional subspaces of $\mathbb{R}^{4}$ what are the possible dimensions of $V\cap W$? I know the answer is: 0, 1, or 2. But I would like more insight ...
1
vote
0answers
68 views

Simple proof that a $3\times 3$-matrix with entries $s$ or $s+1$ cannot have determinant $\pm 1$, if $s>1$.

Let $s>1$ and $A$ be a $3\times 3$ matrix with entries $s$ or $s+1$. Then $\det(A)\ne \pm 1$. The determinant has the form $as+b$ with integers $a$,$b$ and it has to be proven that $a>0$ if ...
1
vote
1answer
31 views

Determinant of a matrix shifted by m

Let $A$ be an $n\times n$ matrix and $Z$ be the $n\times n$ matrix, whose entries are all $m$. Let $S$ be the sum of all the adjoints of $A$. Then my conjecture is $\det(A+Z)=\det(A)+Sm$ , in ...
0
votes
2answers
48 views

How are signs on eigen vectors chosen, am confused? Linear Algebra

I have found the eigen vaues, I also know that you can find the eigenvectors through a Gausian Jordan. -- x1, gauss jordan gives me rows(1 -1/3 ,, 0 0 ), so [a, b] = [1,3] For vector x2, GJ gives ...
0
votes
2answers
32 views

Geometrical interpretation of a solution of a system of linear equations with complex coefficients.

What is the geometrical interpretation of a (unique/infinitely many/no) solution of a system of linear equations of three(or less) equations with three(or less) unknowns with complex coefficients?
1
vote
0answers
32 views

Intersection between 2 lines (3D). This doesn't have a solution does it?

so I was looking through an old exam and this question was given: The teachers answer was the point (9, -9, 21) I tried solving this myself, I got x = x, y = y, but I could not find a point where ...
1
vote
1answer
31 views

Relation on the determinant of a matrix and the product of its diagonal entries?

Let $A$ be a $3\times 3$ symmetric matrix, with three real eigenvalues $\lambda_1,\lambda_2,\lambda_3$, and diagonal entries $a_1,a_2,a_3$, is it true that \begin{equation*} \det ...
1
vote
2answers
54 views

If $\|Tv\|=\|T^*v\|$ for all $v\in V$, then $T$ is a normal operator

I have solved a question but I am not sure the last step of the question. If someone can verify it that would be great. Let $V$ be a finite dimensional vector space with complex inner product. Let ...
0
votes
1answer
17 views

How to find an equation of a plane perpendicular to two other planes and passing through a point

Please, could anybody help me with the next problem. I have two planes: $$ 2x-y+5z+3=0 \ (\text{red plane})\\ x+3y-z-7=0 \ (\text{green plane}) $$ And I need to find a plane which is perpendicular ...
1
vote
0answers
51 views

Proving boundedness of a function .

Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; ...