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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how to complete arbitrary basis knowing 2 orthonormal vectors of Rd (d > 2)

In a paper the following statement is used: "To construct the matrix B, complete the vectors (y, x) to an arbitrary basis of Rd and then apply the Gram-Schmidt orthonormalisation". assume we know x ...
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54 views

How to solve this homogeneous system, with a missing column?

Find the solution set of triplets $(x,y,z)$ that fulfil this system using Gauss-Jordan: $$\begin {cases} -x + 2z = 0\\ 3x - 6z = 0\\2x - 4z = 0\end {cases}$$ First of all, I don't see any ...
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86 views

Obtaining $A$ from $A$$A^t$$A$

Let the matrix $B$=$AA^TA$ be given to us where A is a mxn real matrix.Than how can we obtain $A$ from $B$ ? Can we do the same thing if A is a complex matrix ? I have no idea how to do this ...
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90 views

Dense Countable basis on Hilbert space

Let say that I have a $H$ hilbert space and linear independent countable set $\beta =\{ \beta_1 , \beta_2, \beta_3... \}$ such that $span(\beta)$ is dense set in H. does $span(\beta-\beta_1) =span( ...
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22 views

Characterization of linear transformation with same Kernel and range

Can we characterize class of all linear transformation T:V -> V for which Kernel(T) = Range(T)?
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Using Gauss-Jordan for an infinite-solutions system

I'm starting to get the hang of this Gauss-Jordan stuff - well, I have never done a system with infinite solutions, so I decided to try this one. You can scroll to the bottom instead to see my doubts ...
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104 views

Find all the intersection points of a vector parabola (in R3) and a sphere

Given that I have a vector in R3 (7t, 10t - 2t^2, 5t) | (These numbers are arbitrary for the sake of the process) A sphere centered at the point ( 15, 25, 10) with a radius of 20 There is a ...
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For what Value of K, the following system of equations will have No Solution?

2x-8y = 3 Kx+4y = 10 i can use trial and hit method but its inconvenient and time-consuming, if there is any alternative methods to obtain a solution for these sort of problems, please Let me know. ...
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49 views

Changing to a standard basis

I'm looking at an example problem in my textbook and I'm lost as to how they got the standard basis. $B = \{1,x,x^2\}$ and $C = \{1+x, x+x^2, 1+x^2\}$ of p. Then find the coordinate vector of $P(x) ...
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0answers
135 views

Proof of a property of a cofactor matrix.

If $A$ is a matrix with $n\geq2$, prove the following property of its cofactor matrix - $ {cof} (A^t) = ({cof} (A))^t$. Are the following properties of matrices and determinants of use here - (a) $ ...
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204 views

Finding the dimension of the vector space V and give a basis for V

V = {p(x) in P2 : xP'(x) = P(x)} I let P(x) = a + bx + cx^2 Taking the derivative of this I get P'(x) = b + 2cx However then xP'(x) does not equal P(x) So, do I basically have to keep guessing for ...
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4answers
76 views

Are there no solutions for $\begin {cases} 2x+4y = 6\\ 3x+6y = 5\end {cases}$?

I'm trying to solve an equation system using Gauss-Jordan. $$\begin {cases} 2x+4y = 6\\ 3x+6y = 5\end {cases}$$ So, first, the augmented matrix: \begin{bmatrix} 2&4&5\\ 3&6&6\\ ...
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1answer
296 views

Can all equation systems be reduced to the identity matrix?

I'm trying to learn about solving equation systems using the Gauss-Jordan method. So, you have to convert the equation system to a matrix, and then reduce it to the identity. When you transform it to ...
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1answer
25 views

index of nilpotency of an endomorphism

Let us assume that $E$ is a vector space over $\mathbb R$ or $\mathbb C$ of finite dimension $n$ and $f \in L(E)$ (endormphism over $E$ that is linear maps from $E$ to $E$) such that for all $x\in ...
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2answers
213 views

Prove Derivative is sum of determinants

Given $n^2$ functions $f_{ij}$, each differentiable on an interval (a,b), define $F(x) = det[f_{ij}(x)]$ for each $x$ in $(a,b)$. Prove that the derivative $F'(x)$ is the sum of the determinants, $$ ...
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1answer
825 views

Let $S$ be the subspace of $\mathbb ℝ^3$ spanned by $\bf u$ and $\bf v$. Find the closest point $p$ in $S$ to the point $w$.

Let $S$ be the subspace of $\mathbb ℝ^3$ spanned by vectors $\bf u$ and $\bf v$. Find the closest point $p$ in $S$ to the point $w$, given: $\bf u^T$ = $[1,-2,2]$ $\bf v^T$ = $[-4,-7,-5]$ $w^T$ = ...
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2answers
96 views

Calculate determinant of Vandermonde using specified steps.

$V_n(a_1,a_2\dots, a_n)$ is a $n\times n$ Vandermonde matrix = $$\left|\begin{array}[cccc] 11&a_1&\cdots&a^{n-1}_1\\ 1&a_2&\cdots&a^{n-1}_2\\ ...
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32 views

Question regarding notation involving vector spaces.

Let V be the set of all ordered pairs of real numbers, and consider the following addition and scalar multiplication operations on $u+v=(u_1+v_1+1,u_2+v_2+1),$ $ku=(ku_1,ku_2)$. Show that ...
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50 views

Does a basis of three vectors always span $R^3$

This may be a dumb question, but if I have three linearly independente vectors in $R^3$, will it always span $R^3$ I'm asking this because it's hard to visualize this for every vector. I can imagine ...
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34 views

Need help finding the range(A)

I am having trouble remembering how to find the range. Any help would be great Thanks Let $ \mathbf{A} \ = \ \left(\begin{array}{cc}1&-1&1\\0&1&2\\-1&2&2 \end{array}\right) $ ...
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1answer
66 views

An argument for the Rank-Plus-Nullity Theorem

Could you please comment on the legibility of my proof of the Rank-Plus-Nullity Theorem? Theorem: $A$ is an $(m \times n)$-matrix with $k$ pivot columns. The dimension of the null space of $A$ is $n ...
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2answers
137 views

#10 on GRE Form 8767

In order to send an undetected message to an agent in the field, each letter in the message is replaced by the number of its position in the alphabet and that number is entered in a matrix M. Thus, ...
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1answer
48 views

Solving a simple system of linear equations with a constant

For which values of the constant, c, does the following system have one solution? $ 2x-y=10 $ $ -cx+2y=5 $ The answer, apparently, is that there is a solution for any value of c, given by $ x = ...
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1answer
18 views

How do I prove a component of a vector is 0 for all i in I?

Let $E_1 = (1, 0, \ldots ,0)$, $E_2 = (0, 1, 0, \ldots ,0)$, ... , $E_n = (0, ... ,0, 1)$ be the standard unit vectors of $R^n$. Let $x_1, \ldots ,x_n$ be numbers. What is $x_1E_1 + \ldots + x_nE_n$? ...
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1answer
84 views

Why is $ det(A - \lambda I) = (-1)^n \cdot [\lambda^n + c_1\lambda^{n-1} + … + c_n ] $?

Well the title tells you everything I want to know. Why is $ \det(A - \lambda I) = (-1)^n \cdot [\lambda^n + c_1\lambda^{n-1} + ... + c_n ] $ ? With this I then want to show that $ \det(A - \lambda ...
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3answers
2k views

How to compute the nth power of a matrix

How would I compute $ \big ($ $\begin {matrix} -5 & 8 \\ -4 & 7 \\ \end {matrix}$ $\big )$$^5$ Using the relationship between the diagonal matrices and the nth power of a matrix? My ...
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35 views

Find a general formula for $x_k$

Suppose that the sequence $(x_k)$ is defined by $x_0 = 0, x_1 = 6, x_2 = 1$ and $$x_{k+3} = −x_{k+2}+17x_{k+1}−15x_k\quad \text{for }\, k\geq0.$$ Find a general formula for $x_k$. I have this answer ...
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1answer
260 views

A $2\times2$ Matrix inequality

$M,N$ are $2\times2$ real matrices, and $MN=NM$. Then, for any three real numbers $x,y,z$, we have $$4xz\det(xM^2+yMN+zN^2)\geq(4xz-y^2)\big(x\det(M)-z\det(N)\big)^2 $$ some thought: 1). ...
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1answer
2k views

LU Factorisation (4x4 matrix) - most efficient method

I understand how to do LU factorisation but I'm not sure I'm being very efficient. I first find the row echelon form of A, noting the elementary operations $E_i$ in order. $$ E_1E_2...E_nA = U $$ ...
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0answers
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Is there a name for the group of complex matrices with unimodular determinant?

Does the group $$ G = \left\{ A \in \mathbb{C}^{n \times n} : |\det(A)| = 1 \right\} $$ have a name? It obviously contains the unitary group $U(n)$ and the special linear group $SL(n,\mathbb C)$. ...
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1answer
132 views

Constructing an Isomorphism

I need to construct an isomorphism: $T: W \to C^3$ where W is a subspace of $P_3(C)$ I got bases (for W and $C^3$ respectively) $\alpha = \{(ix + 1),\ \ (x^3 + 2x), \ (2ix^2 + ix -1)\}$ ...
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3answers
172 views

Why is a zero vector space a vector space

The first example in my Linear Algebra textbook states this is so by stating the example statisfies axiom 4, which says if there is zero vector, there is an object u in which 0 + u = u + 0 = u, where ...
3
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1answer
64 views

difference between weak* convergence and convergence

I am trying to prove the following: If $X$ is a finite-dimensional space, then for sequences $\left\{x_n\right\}\subseteq X$ and $\left\{f_n^*\right\}\subseteq X^*$, if there exists an $x\in X^*$ ...
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1answer
81 views

Confusing rational numbers

Question: If $$x = \frac{4\sqrt{2}}{\sqrt{2}+1}$$ Then find value of, $$\frac{1}{\sqrt{2}}*(\frac{x+2}{x-2}+\frac{x+2\sqrt{2}}{x - 2\sqrt{2}})$$ My approach: I rationalized the value of $x$ to ...
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2answers
164 views

Vector spaces and intersections

I was thinking of the following problem lately: Suppose $V_1,V_2,V_3,V_4$ are vector subspaces of $\Bbb{R}^4$ of dimension $2$ such that $V_i\cap V_j=\{0\}$ for $i \neq j$. Is it true that we can ...
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1answer
54 views

How many integers in between 1 and $10^7$ are divisible by 3, 5 or 7?

How many integers in between 1 and $10^7$ are divisible by 3, 5 or 7? I try with that the number of integers between 1 and $10^7$, inclusive, which are relatively prime to 63: $$(10^7)/3=3333333$$ ...
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1answer
54 views

Linear least square solution

The Linear least square solution is obtained by solving $XB =y$ and then $B$ is calculated by following equation $$B =(X^TX)^{-1}Xy $$ Why we go for a pseudo inverse instead of taking a inverse ...
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1answer
22 views

A question for matrix in linear algebra

I'm reading a textbook in which there are some sentences I could not understand. I hope to get some help. Let $B$ be a matrix having rank $k$ and $B=[b_1 \quad b_2 \quad \ldots \quad b_k] \in \mathbb ...
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2answers
82 views

Proving $\nabla_A tr(ABA^T C) = CAB + C^T A B^T$

The above equation appears without proof on page 9 (equation 3) of Andrew Ng's notes on Machine Learning I have tried various approaches to prove this to no avail. From the notes it seems that it ...
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2answers
186 views

Determinant of block $n \times n$ matrix

Let det $A = \det(\begin{bmatrix}B& 0\\ 0& I_mI\end{bmatrix})$; $B$ and $D$ are square matrices. $I_m$ is an identity matrix of size $m$. I keep reading that it is obvious that we can view ...
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0answers
23 views

joint density random variables with a set of equations

There are $n$ equations: $f_i(x_1,x_2,...,x_n,e_i)=0$, $i$ from $1$ to $n$, where $e_i$ are independent random variables whose expectations are all $0$. $x_i$ are random variables. Suppose the map $e ...
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1answer
56 views

Show that the set of vectors are bases for $\mathbb{R}^3$

The set of vectors are: S={ (1,6,4) , (2,4,1) , (-1,2,5) } for it to be a base in $\mathbb{R}^3$ it must be linearly independent and span $\mathbb{R}^3$. I ...
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1answer
37 views

showing that a sequence converges in the dual space of a normed vector space

Suppose that $S=\left\{s_\alpha: \alpha \in A\right\}$ is a set of points in a normed vector space $X$ such that $\overline{span}(S)=X$. If $\left\{f_n\right\}$ is a bounded sequence in $X^*$ and ...
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32 views

Generators of Special Linear Matrix ???

This is a simple question, anyone can help: Can one generate this matrix $A_1$ or $A_2$ or $A_3$ from two matrices $B$, $C$ and their inverse ($B^{-1}$, $C^{-1}$): $$ A_1=\begin{pmatrix} 0& ...
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0answers
83 views

Cramer's rule and understanding Area/Volume

I'm having trouble connecting all the ideas we're learning in Linear Algebra. On the one hand, I understand how to find determinants, and therefore expansion factors. I also am fairly certain I have a ...
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1answer
31 views

show annihilator space is subspace of another annihilator space

Let U and V be subspace of a finite-dimensional linear space X. Prove: If $U \subset V$, then $V^0 \subset U^0$. This doesn't seem right, at least i'm not sure how to approach it.
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2answers
22 views

Help with simple algebra equation

I am doing some math to prepare for a test, and $$\left(\frac{a^2}{b^{-1}}\right)^{-3}\cdot\left(\frac{(2b)^3}{a^{-3}}\right)$$ is apparently $8a^{-3}$. When I try to solve it the closest I get is ...
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1answer
75 views

Find kernel and image in $p : -\frac{x}{2} = y = z$

Linear projection $A : \mathbb{R}_3 \to \mathbb{R}_3$ given as the perpendicular projection on the line $p$ $$p : -\frac{x}{2} = y = z$$ How can i find the kernel and image of projection $A$? i ...
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1answer
92 views

Computing the angle between two vectors using the inner product

Question is: Consider $\mathbb{R}^3$ with the inner product $\langle u1,u2\rangle=x_1x_2+3y_1y_2+z_1z_2$ where $u_1=(x_1,y_1,z_1)$ and $u_2=(x_2,y_2,z_2)$ are two vectors in $\mathbb{R}^3$. What is ...
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2answers
84 views

Row reduced matrix $\Leftrightarrow$ vectors (rows) are linearly independent.

Let $A$, a row-reduced matrix (after applying Gaussian elimination). Show that all rows which are different from $V_0$ (zero vector), are linearly independent. We learned this as sort of an ...