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 ...

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Changing the basis of a transformation

Given $T: \mathbb{R}^2 \to \mathbb{R}^2 : T\begin{bmatrix} x \\ y \end{bmatrix} \to \begin{bmatrix} 2x+y \\ x-3y \end{bmatrix}$ with standard basis $\mathcal{B}$ and basis ...
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29 views

$V=ker\ (T)\oplus im\ (T)$ if $T$ is a self-adjoint

Let $V \;$ be an inner product space over a field $\Bbb{F}$ and let $T:V\to V$ be a self-adjoint linear map. Prove that $V = \operatorname{ker}(T)\oplus\operatorname{im}(T)$. All I can think of is ...
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Tensor algebra becomes an R-algebra. Theorem, Proof, Dummit and Foote

I have the definition of tensor algebra as follows: $T(M) = \bigoplus_{i =1}^{\infty} T^i(M)$, where $M$ is an $R$ module, where $R$ is commutative and contains the element $1$. Finally $T^k(M) = ...
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Solution verification dimension of a union of two sets

Let $V$ be a vector space. Let $W_1$ and $W_2$ be subspace of $V$. Then $W_1+W_2 = \{a+b~|~a\in W_1 \text{ and } b\in W_2\}$. (i)Prove that $W_1+W_2$ is a subspace of V. To show its a ...
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Find vectors when added up equal (1, 1, 1)

Question: Let $V$ be the 2-dim subspace of $R^3$ 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 $V^\perp$, which ...
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About polynomials. If there is not two plynomials with the same grad in S, then S is linearly independent.

The problem states as follows. Let S be a set of polynomials non zero over a field F. If there is not two plynomials with the same grad in S, then S is linearly independent. I tried the following. ...
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computing characteristic polynomial of hyperplane arrangement

The following problem comes from Richard Stanley's $\textit{Enumerative Combinatorics}$ vol. 1, 2nd ed. It is problem 114 (c) in Chapter 3. Let $\mathcal{A}$ be a hyperplane arrangement in ...
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23 views

Area of a parallelogram with three dimensional vectors

There is a parallelogram that has the vertices 0, a, b, and a+b, all of which are three dimensional vectors. a = \begin{pmatrix} 2 \\ -6 \\ 5 \end{pmatrix}b = \begin{pmatrix} -1 \\ -2 \\ 0 ...
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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 ...
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LU growth factor applied to LDL of a Positive Semidefinite matrix

For a Positive Semidefinite matrix $A$, which we can decompose through $LDL$ decomposition as follows: $A=LDL^\text{T}$; how can we prove that for a decomposition $A=LU=L(DL^\text{T})$, the growth ...
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35 views

Square Root Confusion

well we know that $$\sqrt{x^2} = \pm x$$ Then if $$x^2=y^2$$ then $$\pm x= \pm y$$ Does this mean $x = y$ or $-x = -y$ or $x = -y$ or $-x = y$ or all is true? Which is true among these?
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Linear Programming problem

Find values of the variables x1, x2,, and x3 which satisfy x1 + 2x2 + x3 ≤ 16 4x1 + x2 + 3x3 ≤ 30 x1 + 4x2 + 5x3 ≤ 40 so that the minimum value of x1, x2, and x3 is as large as possible. Write this ...
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If zero is an eigenvalue of the matrix A, What does it say about A? [on hold]

An eigenvector cannot be zero, but an eigenvalue can. Suppose that zero is an eigenvalue of A. What does it say about A? Hint: One of the most important properties of a matrix is whether or not it is ...
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2answers
21 views

subspace and subspace perp dimensions equal to V

I'm trying to prove the following statement: if E is a subspace of V, then dim E + dim $E^{\perp}$ = dim V. I know this is true because when these two subspaces are added, they are equal to V, but I'm ...
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Given a survival rate matrix, describe what can be said about it

Given this matrix equation: $$\begin{bmatrix} c_{k+1} \\ t_{k+1} \\ a_{k+1} \\ \end{bmatrix} = \begin{bmatrix} 0 & 0 & 0.33 \\ 0.18 ...
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16 views

Diagonal of multidimensional DFT

If $X$ is a $n\times n$ square matrix and $F$ its Discrete Fourier Transform, is there a way to compute the diagonal $(F_{1,1},\ldots,F_{n,n})$ without explicitly computing the full DFT? How about ...
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22 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 ...
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29 views

finding column vectors - linear transformations

$L:\mathbb{R}^3\rightarrow \mathbb{R}^2$ with bases $\mathcal{S}=\left\{\left(-1,1,0\right),\left(0,1,1\right),\left(1,0,0\right)\right\} \: \text{for} \:\mathbb{R}^3 \:\text{and} \\ ...
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linear transformation of orthogonal vector space on subspace

Let $V$ be a finite dimensional inner product space over $F$. If $W$ is a subspace of $V$, prove that the orthogonal projection of $V$ on $W$ is an idempotent linear transformation of $V$ into $W$. I ...
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1answer
21 views

Calculating determinant matrix with size of n

we got the following matrix in order of $n$x$n$: $$\begin{pmatrix} 1 & 0 & . & . & . & 0 & 1\\ 1 & 1 & 0 & . & . & . & 0\\ 0 & 1 & 1 & 0 ...
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Domain of compostions of linear mappings [on hold]

Let $T$ be a linear transformation from $\Bbb R^3$ into $\Bbb R^2$ and $S$ be a linear transformation from $\Bbb R^2$ into $\Bbb R^3$. Is the mapping $ST$ a linear transformation from $\Bbb R^3$ into ...
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Maps that preserve tensor rank

Suppose we have some tensor product of vector spaces. By tensor rank, I mean the minimal number of simple tensors required to write down an element of this tensor product of spaces. Is there much ...
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Dont ask - What is the relation between $f$ and $f(x)$? [on hold]

$f = a.g + b.h$, space $V$ $f(x) = a.g(x) + b.h(x)$ $f$, $g$ and $h$ are scalar valued functions. $x$ is a vector in $\Bbb R^1$ and $f$ is a vector in $V$. So $[f(x)]$ is a vector. is $[f(x)] (f) ...
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45 views

How do you solve this circular system of equations in $\mathbb{Z}_2$?

I'm trying to solve a system of equations in $\mathbb{Z}_2$ that look like this: \begin{align} x_1 + x_2 = p_1 \\ x_2 + x_3 = p_2 \\ x_3 + x_4 = p_3 \\ ... \\ x_n + x_1 = p_n \\ \end{align} I know ...
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2answers
17 views

How to find matrix of orthogonal projection from gram-schmidt orthogonalization

I'm having a little difficulty understanding Gram-Schmidt orthogonalization. I have a problem to apply Gram-Schmidt orthogonalization to the system of vectors $(1,1,1)^T, (1,2,1)^T$ then write the ...
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1answer
13 views

Showing something involving integrals is an inner product

I have this problem: Let $C([0,1])$ be the real vector space of continuous functions on the interval [0,1]. Show that $<. , .>: C([0,1]) \times C([0,1]) \rightarrow \mathbb{R}$ ...
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1answer
17 views

What the limit of a matrix over time shows about the future

$x_k$ is the fraction of people who prefer cake to pie at year $k$. The remaining fraction $y_k=1-x_k$ prefer pie. At year $k+1$, $\frac{1}{5}$ of those who prefer cake change their mind. Also at year ...
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Matrix with eigenvalue that should equal 1.

I have the matrix: $$A = \begin{bmatrix}4 & -2 & 3\\0 & -1 & 3\\-1 & 2 & -2 \end{bmatrix}$$ and I need to find out if $\lambda = 1$ is an eigenvalue. So I solved the equation ...
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A thought about transition matricies in vector spaces

I am trying to work out the relationship between transformation matricies of a vector space with different bases. I came up with an equation which does not look right, but I would like your opinion. ...
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4answers
43 views

A real $2 \times 2 $ matrix $M$ such that $M^2 = \tiny \begin{pmatrix} -1&0 \\ 0&-1-\epsilon \\ \end{pmatrix}$ , then :

A real $2 \times 2 $ matrix $M$ such that $$M^2 = \begin{pmatrix} -1&0 \\ 0&-1-\epsilon \\ \end{pmatrix}$$ (a) exists for all $\epsilon > 0$. (b) does not exist for any ...
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1answer
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How can I translate this problem into Matrix/Linear Algebra notation?

I have a matrix H of size s×d with holdings of s stocks across d days. H shows how many shares of each stock is in my portfolio on each day. The number of shares can change from day to day due to ...
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1answer
20 views

Congruence Property of Monotone Operators

A map $T$ is called strictly monotone if for $x\ne y$, $\langle u-v,x-y\rangle>0$ for all $u\in T(x),v\in T(y)$. Let $A$ be an $m\times n$ matrix and $b\in\mathbb R^m$. I want to prove that if ...
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1answer
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Finding orthonormal basis using orthogonal basis

I am very confused how to go about finding an orthonormal basis using a orthogonal basis. My book says to just normalize the vectors but it doesnt further explain. When i look at answers for ...
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How can I prove that any matrix A can be expressed as the sum of two Hermitian matrices , B and C, in the form A = B + iC?

The question is in the title really. Whether or not A must also be Hermitian is not clear to me. Sorry, I am not very good with proofs of this nature.
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1answer
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Proving multilinearity of determinant [on hold]

As the title says, how we can prove multilinearity property of determinants: $$ \begin{vmatrix} p+q+r & x+y+z & u+v+w \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33}\\ ...
2
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1answer
41 views

Characteristic polynomial of a matrix polynomial

Thanks for any help or comments. Suppose $A\in M_n(F)$ is an $n\times n$ matrix such that $F$ is a finite field. Also suppose that characteristic polyomial of $A$ is irreducible and is equal to its ...
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1answer
50 views

If $A$ is positive definite then so is $A^k$

Prove that if $A$ is positive definite, then so are $A^2,A^3,\ldots$ and $A^{-1},A^{-2},\ldots$ I know how to show the inverse of positive definite is positive definite but I don't know how to expand ...
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3answers
28 views

Proving $AD_1A^{-1}=D_2$

I want to prove that if $A$ is a permutation matrix, and $D_1$ is diagonal, than $AD_1A^{-1}=D_2$ where $D_2$ is also a diagonal matrix. I have worked out that $A^{-1}=A^T$ and I can see that the ...
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1answer
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linear transformation question (vector) [on hold]

Not sure how to answer this question, please help!
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Different Inverse Approach

As it is known, we use inverse (Gauss Elm, Jordan...) or pseudo-inverse methods (LU, SVD, Chol, QR...) to solve linear equation namely $ A*x=b$ when $A$ is $[m,n]$ and $b$ is $[1,n]$ matrix. These all ...
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centralizers of $X= (M_2(F_p),+)$ in $H=GL_2(F_p)$

I need to find the centralizers of $X= (M_2(F_p),+)$ in $H=GL_2(F_p)$ in order to find the action of $H$ on $X$ which will help me find the orbits of $X$ I Know that the centralizers of $M_2(F_p)$ ...
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15 views

norm of symmetric positive definite matrix

How to prove this? I tried by using the fact that positive definite matrix is diagonalizable by orthogonal matrix and it preserve two norm. But I think this way dosen`t work.
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Find a and b such that the matrix is diagonalizable

Find a and b such that the matrix $$ \left( \begin{array}{ccc} 1 & a \\ 0 & b \\ \end{array} \right) $$ is diagonalizable. I know that $$ D = S^{-1} A S $$ where S is a matrix made of the ...
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1answer
25 views

Condition of distinct eigenvectors?

I am looking at this wikipedia page http://en.wikipedia.org/wiki/Matrix_decomposition#Eigendecomposition ...
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47 views

Vector spaces whose elements are functions

I'm trying to understand what a vector of functions is, from trying to understand how to solve linear homogeneous differential equations. It seems that functions can be manipulated as vectors as ...
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1answer
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error in approximation a monotonic function in L^1

I am trying to solve this problem, but I am getting an incorrect solution. Here is the problem and my approach: Find the best $L^1$ linear approximation of $e^x$ on [0,1] i.e. minimize ...
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27 views

Linear transforms of functions [on hold]

if y is a vector; y = a.sin(x) + b.cos(x), a and b scalar then how is it that for any value of x, y is always a scalar value? Does this mean sin(x) is a vector, and sin(45) is not?
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2answers
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Find value of k for distinct eigenvalues

Consider the matrix $$ A = \left( \begin{array}{ccc} 0 & 1 & 0 \\ 0 & 0 & 1 \\ k & 3 & 0 \end{array} \right) $$ where k is an arbitrary constant. For which values of k does A ...
2
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1answer
38 views

Use determinants to calculate the area bounded by 3 vectors

I have seen the proof of why the area of the parallelogram created by 2 vectors $u = \left(\begin{matrix} u_1\\ u_2 \end{matrix}\right)$ and $v = \left(\begin{matrix}v_1 \\ v_2 \end{matrix}\right)$ ...
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Does this linear system of 5 unknowns and 2 equations have multiple solutions? [on hold]

\begin{cases} x+ 2y - z + w - t = 0 \\ x - y + z + 3w - 2t = 0 \end{cases} Add 1st to the 2nd: $$2x + y + 4w - 3t = 0 \\ y = -2x - 4w + 3t = 0$$ Substitute y in the 1st: $$x - 4x - 8w + 6t - z ...