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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finding a locus in a $3$ dimensions

Given a Tetrahedron $OABC$ such that $O(0,0,0),A(a,0,0),B(0,b,0),C(0,0,c)$ ; $a,b,c$ are not zero. We build a plane $\pi$ that is parallel to $z$-axis and also to $AB$. Plane $\pi$ cuts plane $ABC$ ...
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58 views

Soft Question about Möbius Transformations

Very soft question and I may be completely wrong about this, but does it make any sense to think about the Möbius transformation matrix as a change of basis for $\mathbb C$?
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77 views

Inner-product question

Let $V$ be $\mathbf{R}^2$ equipped with usual inner product, and $v$ be a nonzero vector. $S_v(u)= u- 2 \frac{\langle u,v\rangle}{\langle v,v\rangle } v$ and $\Phi$ be a non-empty set of unit vectors ...
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146 views

Completing two vectors into a basis of $\mathbb R^4$.

Let $e_1=(1,1,0,0)$ and $e_2=(1,1,0,1)$ and $e_3=(0,0,0,1)$. Let $E=\operatorname{span}(e_1,e_2,e_3)$. I want to determine the dimension of the vector subspace $E$. And then complete into a basis of ...
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How do I show a vector in a vector space is unique?

$U = \{(x, y, 0) \in F^3 : x, y ∈ F\}$ and $W = \{(0, 0, z) ∈ F^3 : z ∈ F\}$. Verify that $F^3 = U \oplus W$. Suppose I don't know that if $U$ and $W$ are subspaces of $V$, then $V = U \oplus W$ ...
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96 views

What does it mean for $AA^T$ to be symmetric?

What does it mean for $AA^T$ to be symmetric? A question in my book says to show that $AA^T$ is symmetric so I took a very simple matrix to try and understand this: $A=\begin{bmatrix} 2 \\ 8 \\ ...
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is it possible to generate a unique number given a set of N integers regardless of their permutation?

I need to efficiently compute an "id" for a set of N integers, the id needs to be unique if any of the numbers is different from some other set. At the same time the id needs to be the same if the ...
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82 views

Can anyone explain this isometry to me? $T: (\mathbb R^n,\|\cdot\|_1)^*\to (\mathbb R^n,\|\cdot\|_{\infty})$, $ T(x)(y) = \sum_{i=1}^n x_i y_i$

Can anyone explain this isometry to me? $$T: (\mathbb R^n,\|\cdot\|_1)^*\to (\mathbb R^n,\|\cdot\|_{\infty}),\qquad T(x)(y) = \sum_{i=1}^n x_i y_i$$ I don't get what the domain and image of $T$ are. ...
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to prove $f(P^{-1}AP)=P^{-1}f(A)P$ for an $n\times{n}$ square matrix?

let $f(X)$ be a polynomial and let $A$ be $n\times n$ matrix.We have to show that for any $n\times n$ invertible matrix $P$, $f(P^{-1}AP)=P^{-1}f(A)P$ and that there exist a unitary matrix $U$ such ...
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35 views

Constructing a basis for $\mathbb R_n[X]$

Let $\mathbb R_n[X]$ be the vector space of reel polynomials of degree at most $n$. Let $P_0, \cdots, P_n$ be a family of polynomials such that $deg(P_i)=i$ for all $i=0..n$. Show that this family is ...
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47 views

linear transformation relative to a basis

I am self-learning linear algebra and I get stick on the Linear transformation from a vector space to itself. My book (linear algebra and its application 4th edition) say that $$[T(x)]_{\beta} = ...
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79 views

Linear Independent proof

In my Linear Algebra class we define Linear dependence as follows: If $F$ is a field and $V$ is a vector space over the field $F$. The set $A = {\lbrace v_1,v_2,...,v_k \rbrace}$ where ...
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310 views

Rank of a Matrix and Echelon Form to determine ranks.

What is the meaning rank of a matrix in terms of vectors, and how does Echelon form work in determining the rank of a matrix?
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49 views

If W is a finite dimensional i.p.s and V is a subspace of W, and if $T \in L(V,W)$ is the map $Tu=u$, how can I prove $T^* = P_{V}$?

If W is an inner product space that is also finite dim. and V is a subspace of W, and if $T \in L(V,W)$ is the map $Tu=u$, how can I prove $T^* = P_{V}$? Here $T^*$ is the adjoint operator of $T$ and ...
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47 views

How to disprove that there is an inner product on $\mathbb{R^2}$ s.t. the norm is $||(x_1,x_2)|| = |x_1|+|x_2|$?

How can I disprove that there is an inner product on $\mathbb{R^2}$ s.t. the norm is $||(x_1,x_2)|| = |x_1|+|x_2|$? My approach is to use the parallelogram law to show that if I have two vectors $u,w ...
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1k views

Basis for set of nxn matrices with trace = 0

I am trying to find a basis for the set of all $n \times n$ matrices with trace $0$. I know that part of that basis will be matrices with $1$ in only one entry and $0$ for all others for entries ...
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117 views

Prove that $T$ is an isomorphism if and only if $T(\beta)$ is a basis for $W$.

Let $V$ and $W$ be $n$-dimensional vector spaces, and let $T:V\rightarrow W$ be a linear transformation. Suppose that $\beta$ is a basis for $V$. Prove that $T$ is an isomorphism if and only if ...
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40 views

Least squares method: must each partial derivative be zero?

In gradient equations, does the sum of the partial derivatives have to be equal to zero or each derivatives has to be zero? As I have just started to understand gradient equations, if my question is ...
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97 views

How can I prove this simple result?

Let $A$ be a $n \times n$ not symmetric matrix where all of its rows sums to zero. How can I show that the linear system $Ax = 0$ has infinite solutions and all of them are $x = k\cdot(1, 1, \ldots, ...
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161 views

Finding a transformation matrix with respect to a coordinate mapping

Determine the transformation matrix of $T$ with respect to the coordinate mapping: $\xi(p) = (p(-1),p(0),p(1))$, and we define $T:\mathcal{P}_2 \to > \mathcal{P}_2$, with ...
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85 views

Definition of “symmetric bilinear (real) form indefinite”

In my studies I use these definition: Def.: $f \in \mathscr{B}_ \Bbb{R}((e \times e), \Bbb{R}) $, $f $ is symmetric bilinear (real) form positive definite if 1) $\forall x \in e(f(x,x)\geq0)$ 2) ...
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44 views

Dimension and its consequences

Find a basis for $S=[\mathrm{(1,2,3),(3,4,7),(5,-2,3)}]\subseteq \Bbb R^3$ and give the dimension. Then, putting all the vectors as the columns of a new matrix: $A=\begin{bmatrix} 1 & 3 & ...
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56 views

Symmetric bilinear form that is not diagonalizable

Given a field $F$ of characteristic 2, I need to give an example of symmetric bilinear form $f$ on a finite dimentional $F$ vector space such that $f$ is not diagonalizable. I will appreciate any ...
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68 views

independent/dependent values at different frequencies and phases

I am curious about the following problem. I would like to ask for help solving it. Consider the following $m$ sinusoidal functions $\sin(\omega_{1}⋅t+\phi_1),\sin(\omega_{2}⋅t+\phi_2),..., ...
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43 views

Do vectors $(\sin n x)_{n=1}^d$ always form a basis of ${\mathbb R}^d$ for $d$ distinct $x \in (0, \pi )$?

Points on the moment curve $(y,y^2,\ldots, y^d)$ always form a basis of ${\mathbb R}^d$ for $d$ different non-zero choices of $y$. The determinant formula for a Vandermonde matrix says that this is ...
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43 views

Determining if a subset is a subspace

If I have the following: $span{(1, 1, 0, 1)} \cup span{(1, 0, 1, 1)}$ How can I determine if this subset of $R^4$ is a subspace of $R^4$?
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178 views

Can a matrix be just 1 number?

Can a matrix be just one number? Eg. Does 2 count as a matrix, if the question asks for a matrix to fit the question, but only the number 2 multiplying this particular matrix gives me the desired ...
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1answer
66 views

Mathematical Induction on Matrix Sequence

Given $M_0=(1)_{1\times1}$. Denote: $$ M_{i+1}=\begin{pmatrix} M_i &-M_i \\ M_i & M_i \end{pmatrix} \; for\; i=0,1,2,...$$ Prove that $M_i$ is a square matrix of order $2^i \times2^i$ and ...
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78 views

1040 grid, 3 boxed, what space?

A grid I have a grid with boxes. The total grid is 1040 wide. Number of boxes and spaces I have 3 boxes an 2 spaces between them. Question What are the number of pixels between the boxes, the ...
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61 views

Different basis over $\Bbb{R}$ and $\Bbb{C}$

V is a finite dimensional vector space over $\Bbb{C}$ and {v$_1$,...,v$_n$} be a basis of V. Show {v$_1$,iv$_1$,...,v$_n$,iv$_n$} is a basis of V over $\Bbb{R}$ and conclude: ...
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Holder inequality for matrices

I am interested in the following version of the Holder inequality. Let $D \in M_n(\mathbb{C})$ be a positive semi-definite matrix of trace $1$ and $A, B \in M_n(\mathbb{C}).$ Does it follow that $$ ...
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57 views

Equivalent definitions of isometry

Consider a map $T:\mathbb{R}^2\to\mathbb{R}^2$ such that $\lVert T(x)\rVert=\lVert x\rVert$. Is this equivalent to stating that $\langle x, y\rangle=\langle T(x), T(y)\rangle$ for all ...
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75 views

For a matrix $A$, is $\|A\| \leq {\lambda}^{1/2}$ true?

In class I saw a proof that went something along these lines: Define $\|A\| = \sup \dfrac{\|Av\|}{\|v\|}$ for v in V, where the norm used is the standard (Does this even exist?) Euclidean norm in V. ...
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52 views

Extending a set of vectors to a basis by picking from a given basis

I have a linear independent set ${\cal K}=\{v_1,\dots,v_{k-d}\}\subset\mathbb{R}^k$. I'd like to find $\cal W=\{w_1,\dots,w_d\}$ such that $\cal K\cup W$ is a basis for $\mathbb{R}^k$. To do this, ...
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42 views

Properties of derived group

If K is a normal subgroup of G then does the following equality hold: $[G/K,G/K]=[G,G]/(K \cap[G,G]) $ If this is true then prove it and if not then give a counter example.
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66 views

Explanation of: The vector $(\alpha, \beta)$ is parallel to the line $Ax + By + C = 0$ if and only if $A\alpha +B\beta = 0$

I just stumbled on this passus in my textbook and I cannot really make sense of it: The vector $(\alpha, \beta)$ is parallel to the line $Ax + By + C = 0$ if and only if $A\alpha +B\beta = 0$. ...
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1answer
24 views

Cardinality of the intersection between an open set and an affine linear subspace

I'm trying to prove the following: Let $S \subseteq \mathbb{R}^N$ be an affine linear subspace with $\dim(S) \geq 1$, and $A = (0,1)^N \subset \mathbb{R}^N$. Suppose that there exists a vector $x \in ...
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52 views

Local coordinates for two riemannian metrics

Let $(M,g)$ be a Riemannian manifold, $g' = g + f$ be another metric. Is it possible to get local coordinates such that at a point $P \in M$, $g_{ij} = \delta_{ij}$ and $f_{ij} = 0$ for all $i \not = ...
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139 views

Homogeneity and Differentiability at $0$ implies linearity?

Suppose $f: \mathbb{R}^n \to \mathbb{R}^m$ is homogeneous and differentiable at $0$, then does it follow that $f$ is a linear transformation? I know that I need to show that for any $x,y \in ...
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Finding the inverse of a matrix using a series

I want to find the inverse of the matrix $A$ given by: $ \left( \begin{array}{cc} 1 & -\epsilon \\ \epsilon & 1 \\ \end{array} \right) $ where $|\epsilon|$ $< 1$ (although I do not ...
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Sketch the polar graph r=2+cos(theta). Find the points of intersection, if any, of this graph with the straight line y=2x-1 (use two decimal places)

I have already sketch the polar graph. and I have to find this graph's intersection point with the straight line y=2x-1 so, I try to solve it like this way: y=2x-1 Rsin(theta)=2Rcos(theta)-1 ...
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1answer
132 views

Find an invertible matrix B such that BAB^-1 takes a specific form [closed]

The question is to find an invertible matrix $B$ such that $B^{-1}AB$ making $A$ becomes \begin{pmatrix} a & b & * &* &* \\ -b & a &* &* &* \\ 0& 0 & ...
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Transpose or not before finding basis for subspaces V and U

$U$ is a subspace with basis a and b. $V$ is a subspace with basis c and d. All vectors - a b c d - are 4x1-vectors. Find a basis for $U+V$. I think the algorithm here is to put a b c d inside a ...
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Determining which sets are spanning sets for P3?

I have a list of sets: (a): {$1, x^2, x^2 - 2$} (b): {$2, x^2, x, 2x + 3$} (c): {$x + 2, x + 1, x^2 - 1$} (d): {$x + 2, x^2 -1$} I am supposed to determine which of the following are spanning ...
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389 views

Determining Linear Dependence/Independence of vectors in R2x2?

I have the following vectors and need to determine if they are linearly independent/dependent. I know that for linear independence, the coefficients multiplied by each vector must equal zero, however, ...
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20 views

show all invariant subspaces are of the form

here u_L is minimal polynomial there's a theorem that says: if V = C_x then deg of min poly = dim V.
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32 views

Substitution method for exact differential equation help

I'm trying to use the substitution method to solve for a differential equation. The equation is $y'=(x+y-3)^2 $, $ y(0) = 0$ I used substitution to get $u=x+y-3$ and $u'=1+y'$. So the final ...
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1answer
59 views

Eigenvalues and Corresponding Eigenspace Bases

Could someone describe the eigenvalues of $ \left( \begin{array}{cc} 2 & 1 \\ -1 & 2 \end{array} \right) $, as well as the bases of the corresponding eigenspaces? I received eigenvalues ...
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2answers
198 views

Can two rectangular matrices multiply to yield the Identity?

For example, does there exist a 3x2 matrix and a 2x3 matrix such that the product is the 3x3 Identity matrix? If not, how would you go about proving that?
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1answer
29 views

Show that a Hermitian Matrix has a zero column and row

Given a Hermitian matrix $A$ with largest eigenvalue $\lambda$, can we show that if $\lambda$ lies on the diagonal of $A$ (say at $a_{ii}$) that $a_{ij} = a_{ji} = 0 $ whenever $i \neq j$? If we ...