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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I need help on manipulating this expression:

Assume that $(4k + 3) ^ 2 - (4k + 3)$ is not divisible by 4. If this is true, prove that $(4(k+1) + 3) ^ 2 - (4(k+1) + 3)$ is not divisible by 4. I need to prove this for my induction problem, and ...
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1answer
118 views

Determining if a set is a generating set for $R^n$

I have \begin{bmatrix} -1 & 0 & 3 & -5 \\[0.3em] 1 & -1 & -7 & 7 \\[0.3em] 2 & 2 & 2 & 6 \end{bmatrix} and I need to prove if ...
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2answers
253 views

How to denote the opposite case of the Kronecker Delta?

The Kronecker delta is defined as link to wikipedia: $$\delta_{l,m} = \begin{cases} 1 & \text{if }m=l,\\ 0 & \text{if }m\neq l. \end{cases}$$ I would like to denote the case where: $$ = ...
2
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2answers
367 views

Vector space over complex numbers is also a vector space over the real numbers

So I know I need to prove the kajillion axioms of vector space like commutativity, associativity, the additive/multiplicative identities/inverses etc. How would I go about getting started?
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0answers
96 views

Multi objective optimization into single objective.

I read that it is possible to convert a multi-objective optimization problem into single objective by using weighted sum method. I wanted to know if it is a good idea to convert a two objective ...
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1answer
55 views

Subspaces of a vector

Prove that if $U$ and $W$ are subspaces of vector space $V$ such that $V=U\cup W$ then either $V=U$ or $V=W$. I am thinking one should use bases for $U$ and for $W$ somehow to show that bases for $V$ ...
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1answer
53 views

Diagonalizability of a linear transformation

I'm trying to prove the following: Let $ T: V -> V $ be a linear transformation and let $ \lambda_1 , ... , \lambda_k $ be distinct eigenvalues of T. Suppose the characteristic polynomial of T is ...
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1answer
48 views

About Linear Independence

If a vector in the set of vectors $$\{v_1,v_2,v_3...v_n\}$$ can be written as a linear combination of other vectors, must a row in the matrix formed by linearly combining the vectors be equal to a ...
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1answer
3k views

How do you determine whether a given set of functions is a subspace of C[-1,1]?

I'm having a terrible time understanding subspaces (and, well, linear algebra in general). I'm presented with the problem: Determine whether the following are subspaces of C[-1,1]: a) The set ...
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4answers
61 views

Can an $m \times n$ rank $1$ matrix be written as a product of an $m\times1$ and a $1\times n$ matrix?

If a $m \times n$ matrix has rank $1$, does it imply that it can be written as a product of one $m\times1$ and one $1\times n$ matrix. How to prove it ? Is this decomposition unique ? What are the ...
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1answer
43 views

Simplifying $\mathbf{X}(\mathbf{X}+a\mathbf{I})^{-1}$

I'm having trouble simplifying the following expression in matrix form: $$\mathbf{X}(\mathbf{X}+a\mathbf{I})^{-1}$$ Where $\mathbf{X}$ is an invertible $n \times n$ matrix, $a$ is a scalar value, ...
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2answers
45 views

Describe the subspace $T^{-1}(N)$

I got a) and b), but I have no idea about c).
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2answers
225 views

Is matrix multiplication by an invertible matrix one-to-one and onto?

Maybe I'm just not very experienced on the nitty gritties of matrix multiplication, but is the function $f(X)=AX$ where $X$ is a square matrix and $A$ is an invertible matrix one-to-one and onto? How ...
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1answer
123 views

How to prove $AB$ is a diagonalizable matrix?

Let $A$ be a positive definite matrix, $B$ an Hermitian matrix. How to prove $AB$ is a diagonalizable matrix?
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2answers
143 views

Is it possible to use the imaginary components of quaternions to facilitate calculation of vector cross products?

It has come to my attention that the cross products of the vectors $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ are almost identical to the products of the imaginary components of quaternions $i$, ...
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0answers
26 views

If $B_A\cup B_B$ is a basis for $E$, then not necessarely $E=A\oplus B$.

Consider the vector space $\mathbb R^3$ and an endomorphism $f:\mathbb R^3\rightarrow \mathbb R^3$. Suppose we are given $A$ the matrix of $f$ relative to the canonical bases, and from which we ...
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1answer
201 views

Quadratic form as a ratio of determinants

I am looking for some hints to prove the following equality: $y^{\top}y - y^{\top}X(X^{\top}X)^{-1}X^{\top}y = \dfrac{\det(L^{\top}L)}{\det(X^{\top}X)},$ where $y$ is a $n\times 1$ vector, $X$ is a ...
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1answer
44 views

Properties for internal stability of a discrete-time system

These are two parts of a larger proof I'm working on, can't figure how i) implies ii) though. Dynamic system: $x_{(k+1)} = Ax_{k}, x(0)=x_0$ Where $A \in \mathbb{R}^{n\times n} $ is a real ...
3
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2answers
809 views

Characteristic polynomial of an inverse

Given the characteristic polynomial $\chi_A$ of an invertible matrix $A$, I'm to find $\chi_{A^{-1}}$. I can see that this is theoretically possible. $\chi_A$ uniquely determines the similarity class ...
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2answers
85 views

What is the geometric meaning of the number of independent derivatives of $\gamma$?

Let $\gamma:I \to \mathbb{R}^n$ be a curve. I want to see, what is a geometric meaning of the number of independent derivatives of $\gamma$. I guessed it is it's dimension but it was not. Can you help ...
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2answers
203 views

If $A^2\succ B^2$, then necessarily $A\succ B$

I remember reading somewhere about the following properties of non-negative definite matrix. But I don't know how to prove it now. Let $A$ and $B$ be two non-negative definite matrices. If $A^2\succ ...
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1answer
55 views

can we decompose $\mathcal{A}$ of $\Bbb{R}^n$as an orthogonal transformation and a dilation?

Problem Any linear transformation of $\Bbb{R}^n$ is the composition of an orthogonal transformation and a dilation along perpendicular directions(with distinct coefficients) for any linear ...
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1answer
94 views

Direct sum and subspaces

The question states, "Find subspaces W, X, Y ⊂ ℝ2 with ℝ2=W⊕X=W⊕Y, but X does not equal Y." So if W and X are the Axis in R2 and Y can not equal X, how do you get W and Y to equal R2 without Y ...
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2answers
101 views

Transfomation of one coordinate system to a another

I have a molecule with one coordinate system ( denote as x,y,z ) where the origin is center of mass of the molecule. I have to define another coordinate system (p,q,r) for a local motion. (shown in ...
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6answers
2k views

Cool mathematics I can show to calculus students.

I am a TA for theoretical linear algebra and calculus course this semester. This is an advanced course for strong freshmen. Every discussion section I am trying to show my students (give them as a ...
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1answer
288 views

Subspaces of finite fields viewed as vector spaces on itself

How can I find the number of linear subspaces of dimensions 1 and 2 of the n- dimensional vector space $\mathbb{Z}^n_p$ over the field $\mathbb{Z}_p$?
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5answers
225 views

Why isn't removing zero rows an elementary operation?

My prof taught us that during Gaussian Elimination, we can perform three elementary operations to transform the matrix: 1) Multiple both sides of a row by a non-zero constant 2) Add or subtract rows ...
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2answers
84 views

Why are $R^n$ treated as $R^{n+1}$ spaces in $R^{n+1}$?

$y = x$ is a line in $R^2$ space. But if you graph $z = x$ in $R^3$ space, it's a plane: Both functions have the same relations, so why is one a plane but the other a line?
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2answers
21 views

if $v$ is a member of $H$ and $v$ is not a member of $M$ then $u$ is member of $K$. How is this possible?

Let $(V,K)$ and $u,v$ is a member of $V$. Suppose that $M$ is a subset of $V$ is a subspace of $V$ with basis $B_m=\{m_1,...,m_r\}$ with $r$ less than and equal to $n$. Let $H$ be a subspace spanned ...
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2answers
3k views

Find point on line closest to another given point.

Find the point on the line $x=[1,1,1]+t[1,2,3],\ t \in \mathbb{R}$, that is closest to the point $[0,0,1]$. How do you find this point? Thank you very much.
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1answer
82 views

Norm of the product of an arbitrary matrix and an orthogonal matrix.

Prove the following directly. (Do not use the fact that relates $\left\|A\right\|_2$ to the maximum eigenvalue of $A^{T}A$.) (a) (I've got this one already) If $D$ is the $n\times n$ diagonal matrix ...
2
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2answers
181 views

Find quotient group $GL(n, \mathbb{C})/H$, where $H$ is a group of invertible matrices $GL(n, \mathbb{C})$ with $det \in \mathbb{R}$.

Find quotient (factor) group $GL(n, \mathbb{C})/H$, where $H$ is a group of invertible matrices $GL(n, \mathbb{C})$ with $det \in \mathbb{R}$. I suppose, the theorem that for any group homomorphism ...
2
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1answer
150 views

Find Orbit of element $m \in M, M = M(2, \mathbb{R})$ under action of group $G = GL(2, \mathbb{R})$ mapping $m$ to $g^{-1}mg$, $g \in G$.

Find Orbit of element $m \in M, M = M(2, \mathbb{R})$ under action of group $G = GL(2, \mathbb{R})$ mapping $m$ to $g^{-1}mg$, $g \in G$. The element $m$ is $ \left( \begin{matrix} 2 & 1 \\ 0 ...
3
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1answer
222 views

Criterion for positive semi-definite matrices

Could anyone give me hint: is there a criterion for positive semi-definiteness of a matrix, in terms of dimension reduction, i.e, such that positive semi-definiteness of $n\times n$ matrix is ...
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0answers
133 views

Solve system of matrix equations in finite field

I have the following system of matrix equations: \[ X_1 = A_1 X_2 B_1, \] \[ X_2 = A_2 X_1 B_2; \] where $A_i$, $B_i$ and $X_i$ are $n\times n$ matrices (for even $n$) over a finite field ...
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2answers
39 views

Finding a system of linear equations which result is a given matrice

Say I have three matrices forming a true equation as follows: $$ M= \begin{bmatrix} 1 & 3 & 1\\ -2 & 2 & 6\\ 1 & 1 & 1\\ \end{bmatrix} \times \begin{bmatrix} ...
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1answer
181 views

Can the measurement matrix used for compressive sensing be a sparse matrix?

I am interested in analyzing Compressive Mechanism: Utilizing Sparse Representation in Differential Privacy. In my research, the measurement matrix $A\mathbb \in R^{m \times n}$ needs to be sparse. ...
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1answer
29 views

If a vector $v \in C(A)^\perp$, does that mean $v \in N(A)$ or that $v \in N(A^T)$ and why?

...And what is the difference between $N(A)$ and $N(A^T)$? I'm trying to understand the difference between perpendicular projection matrices and the span of the null space of a subspace. Could you ...
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1answer
60 views

Show set is a basis - complex sequences which are eventually zero

Let $c_{00}$ be the subspace of all sequences of complex numbers that are "eventually zero". i.e. for an element $x \in c_{00}$, $\exists N \in \mathbb N$ such that $x_n =0, \forall n\ge n$. Let ...
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1answer
34 views

Determine a linear application

How can I determine a linear application $f: \mathbb{R}^2 \to \mathbb{R}^2 $ such that $Kerf = span[ (-1, -5) $? By the definition of Ker, we know that $ kerf = [ (x,y) \in \mathbb{R}^2 : f(x,y) = ...
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2answers
122 views

Show that there exists a vector $y$ such that every $x \in X$ can be written uniquely in the form $x=\lambda y+z$

Let $f$ be a linear functional on the vector space $X$ over $F$, $f \neq 0$, and $N$ is the nullspace of $f$. Show that there exists a vector $y$ such that every $x \in X$ can be written uniquely in ...
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1answer
539 views

Linear Transformation: P2(R) -> P3(R)

I have to verify the dimension formula for this: $T: P2(R)->P3(R) $ defined by $T(f(x))=xf(x)+f'(x)$ I have worked out that the null space of T is when f(x) is = 0. But isn't the range all of ...
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1answer
42 views

property of the cross product and dot product

I have this exercise that is difficult for me. $a, b, c$ are three $3$-vectors and $$(a+b-c)\cdot(a-b+c) \wedge (-a+b+c) = -4a\cdot b\wedge c$$ where $\cdot$ is the dot product, $\wedge$ is cross ...
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1answer
339 views

Proving linear independence of infinite set (monomials)

I would like to prove that the set of monomials is linearly independent in the complex linear space $C(\mathbb R)$. I understand the definition of linear independence and I'm stuck on how to prove ...
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1answer
111 views

linearly independent commuting $2\times 2$ complex matrices (Hoffman Kunzze, Linear algebra, 6.5.2)

Actual Question is: Let $\mathcal{F}$ be a commuting family of $3\times 3$ complex matrices. How many linearly independent matrices can $\mathcal{F}$ contain? what about the $n\times n$ case? ...
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8answers
204 views

Systems of linear equations to calculate $\alpha$ and $\beta$

Point $1$: When there is $1$ car passing the road, the average speed is $50$ km/h. Point $2$: When there are $5$ cars passing the road, the average speed is $45$ km/h. Point $3$: When there are $12$ ...
1
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1answer
53 views

Fit image to maximum resolution size

I have a maximum number of pixels that I can process in my app at one time. The user must be able to open any image and if it's bigger than the maximum number of pixels it should be scaled down to ...
3
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0answers
28 views

LU Factorisation - When are there 0s in L and U? (Strang P96 & P105 2.6.21)

Assume no row exchanges. When can we predict zeros in $L$ and $U$ ? $1.$ When a row of $A$ starts with zeros, so does that row of $L$. $2.$ When a column of $A$ starts with zeros, so does ...
3
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1answer
215 views

Uniqueness of LDU Factorisation [Strang P105 2.6.18]

Let $L$ be a lower triangular matrix, $D$ diagonal, and $U$ upper triangular. If $A = LDU$ and also $A = L_1D_1U_1$ with all factors invertible, then $L = L_1$ and $D = D_1$ and $U = U_1$. 'The ...
1
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2answers
707 views

product of m x n matrix with n x m matrix

How to prove that product of $\mathbb{m x n}$ matrix with $\mathbb{n x m}$ matrix is not invertible given $\mathbb{m >n}$. For the case of $\mathbb{2 x 1}$ and $\mathbb{1 x 2}$ matrix, it is ...