-1
votes
1answer
47 views

How to find a transition matrix?

let $a= \{a_1, a_2, a_3, a_4\}$ and $b=\{b_1,......,b_4\}$ and $r = \{r_1,...,r_4\}$ Also, $b_1 = 4a_1$ $b_2 = 8a_1 + 7a_2$ $b_3 = 4a_1 + 4a_2 + 4a_3$ $b_4 = 9a_1 + 5a_2 + 8a_3 + 5a_4$ and ...
0
votes
1answer
12 views

what is the dimension of this subspace for given problem

In a subspace $W=\{[a_{ij}]:a_{ij}=0$ if $i$ is even$\}$ of all $10\times 10$ real matrix, what is the dimension of W?
0
votes
1answer
20 views

Show the Hermitian matrices, with trace(g*g1,1)=0 form a vector space.

This is a question from an example sheet that I think may have a mistake in it. Show that the set of Hermitian matrices $A \in H_2 (\mathbb{C})$ with Trace$(A\cdot A_{(1,1)})=0$ is a real three ...
1
vote
1answer
29 views

$\frac{d(X'X)}{dX}=?$

Thanks a lot for reading my thread. I am wondering what is the derivative of $X'X$ with respect to $X$? Here $X$ is a vector/matrix, and $X'$ is the Hermitian matrix of $X$; It would be great if ...
1
vote
1answer
20 views

sum of two matrices question given condition

How can it be proved that two matrices being orthogonally diagonalizable indicates that their sum is also?
1
vote
0answers
21 views

How to calculate normal (of magnitude 1) of a triangle?

I am currently doing a bit of geometry practice and wanted to know how to calculate the normal (of magnitude 1) of a triangle defined by 3 vertices: a, b and c`. ...
1
vote
0answers
30 views

three dimensional subspace question

If a vector is in $\mathbb{R}^5$, does this mean that the projection of this vector onto $S$ is in $\mathbb{R}^3$, where $S$ is some 3-dim subspace of $\mathbb{R}^5$?
1
vote
2answers
46 views

Consistency of matrix norm: $||Ax||_2 \leq ||A||_{Frobenius}||x||_2$

I'm trying to show that $||Ax||_2 \leq ||A||_{F}||x||_2$ where $A$ is an n by n matrix, $x\in \mathbb R^n$, $||x||_2$ is the euclidean norm, and $||A||_F$ is the frobenius norm. I actually wrote ...
1
vote
2answers
26 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 ...
0
votes
2answers
53 views

Why I need to study Matrix and Vectors in maths

I am presently learning C and C++ programming. I want to make my profession as a C and C++ programmer. Well. In Data structure concepts, I can see lot of matrix material. In school time, I used to ...
-1
votes
1answer
38 views

matrices forms a basis for vector space 2x2

$\begin{bmatrix}0&1\\2&3\end{bmatrix}$ $\begin{bmatrix}3&4\\5&6\end{bmatrix}$ $\begin{bmatrix}7&8\\9&10\end{bmatrix}$ $\begin{bmatrix}11&12\\13&14\end{bmatrix}$ Show ...
0
votes
1answer
30 views

How to show a subset doesn't span a space?

Given that $\{v_1,…,v_m\}$ is linearly independent, how do you show that $\{v_2,…,v_m\}$ does not span that same vector space?
3
votes
2answers
57 views

Finding a basis for a given subspace of $\Bbb R^4$

Find a basis for the subspace $ W = \{(x, y, z, w) \in\Bbb R^4 : y − 2z + w = 0\}$. What is $\dim(W)$? I don't seem to understand how to solve this problem. I just don't know where to start I am not ...
0
votes
1answer
30 views

A trivial solution vs. a non-trivial solution - involving vectors

I'm not entirely sure I understood this question in my text book, but it said the following: The zero vector $0 = \left(0,0,0\right)$ can be written as a linear combination of the vectors $v_1$, ...
1
vote
1answer
60 views

What is transition matrix

Every e_j har coordinates in the first base: $$e_j = \sum_{i} s_{ij}e_i $$ so writing those coordinates as column vectors we get an important transition matrix $S = (s_{ij})$ and Theorem: ...
0
votes
1answer
19 views

Invariant subspaces using matrix of linear operator

I am attempting the following problem but stuck at some parts: How does one find the (2 dimensional) subspaces that are invariant under $A$ for $$A = \begin{pmatrix} 1 & 0 & 0 \\ 0 &2 ...
1
vote
3answers
35 views

Linear Algebra Vector Space matrix help

Let $M_{2\times2}$ be a vector space of all $2\times2$ matrices. If the transformation from $M_{2\times2}$ to $M_{2\times2}$ is $t(A)=A+A^T$ and $A$ is a $2\times2$ matrix with the top row $a,b$ and ...
2
votes
1answer
42 views

Understanding the significance of row space and column space basis

I've just learned about the row and column space basis and I'm confused about what the significance of each is. My professor basically hasn't said much and has danced around any direct questions on ...
3
votes
4answers
79 views

Prove that for every vector $V$, $||V||_{\infty} \leq ||V||_2 \leq || V||_1$

$\newcommand{\inf}{||V||_\infty}$ $\newcommand{\two}{||V||_2}$ $\newcommand{\one}{||V||_1}$ Prove that for every vector $V$, $\inf \leq \two \leq \one$ I have tried to look online for a solution to ...
1
vote
2answers
23 views

Finding column and row space without computing A.

I have the a question that asks that I find the column space and row space of: $$A = \begin{bmatrix}1&2 \\4&5 \\2&7\end{bmatrix} \begin{bmatrix}3&0&3 \\1&1&2 ...
1
vote
1answer
28 views

Are orthogonal spaces exhaustive, i.e. is every vector in either the column space or its orthogonal complement?

Quick question about subspaces, just to make sure I have this straight in my head. Taking an $n\times k$ matrix X with $rank(X)=k$, is every vector in $\mathbb{R}^n$ in either the column space $C(X)$ ...
0
votes
3answers
40 views

how to find a matrix A given the solution?

if we need,for example, to find a nonzero 3x3 matrix A such that we are given a 3x1 vector as a solution to Ax = 0. What is the general procedure we can follow to obtain such Matrix A? Thank you :)
-1
votes
1answer
27 views

For each vector space of functions of one real variable, represent the derivative transformation with respect to B,B.

{acosx +bsinx | a,b ∈ R}, B = < cosx, sinx > My work is the following: {{1},{0}} h-> cos x {{0},{1}} h-> sin x RepB(cos x) = {{1},{0}} since cos x = Acosx +Bsinx RepB(sin x) = {{0},{1}} My ...
0
votes
1answer
32 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, ...
5
votes
3answers
109 views

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 ...
0
votes
1answer
39 views

Stochastic matrix problem

A (left) stochastic matrix is one which has only non-negative entries, and such that the entries in each column sums up to 1. Let $A$ be any (general) 2x2 stochastic matrix. a) Show that one of the ...
1
vote
2answers
69 views

Show that a set of vectors spans $\Bbb R^3$?

Let $ S = \{ (1,1,0), (0,1,1), (1,0,1) \} \subset \Bbb R^3 .$ a) Show that S spans $\Bbb R^3$ b) Show that S is a basis for $\Bbb R^3 $ I cannot use the rank-dimension method for (a). Is it ...
0
votes
2answers
44 views

Vector space and Dual space

I'm struggling with this problem: Let $V$ be a vector space over a field $F$ and let there be $l_1,l_2 \in V^*$. I need to show that if $l_1(x)l_2(x)=0$ for every $x \in V$ then at least one of ...
0
votes
1answer
27 views

Two different ways to write C(A)?

let $\mathrm A \in \Bbb R^{m\times n}$ I know that the three fundamental subspaces are: $\mathrm \ker(\mathrm A) = \{ x \in \Bbb R^n : \mathrm Ax = 0 \} = \{x\in \Bbb R^n : \langle ...
1
vote
1answer
61 views

Getting stonewalled on computation of $2\times 2$ Hessian matrix

The question: Let $z \in R^N$, and let $f(z) = \log[1^T z] \in R$. I am told that the Hessian matrix of this function is the following: $$ H = \frac{1}{1^Tz}\Big[ 1^Tz \mathrm{diag}(z) - zz^T \Big] ...
2
votes
1answer
98 views

Linearly independent functionals

Let $ f_1,\ldots,f_n$ be linearly independent linear functionals on a vector space $X$. Show that there are $n$ elements $x_1,\ldots,x_n$ in $X$ such that the $n\times n$ matrix $[f_i(x_j) ]$ is ...
1
vote
1answer
46 views

Relation between condition and linear dependence of column vectors

There are several interpretations of the condition number of a matrix: Relation between smallest and largest singular values, amount of error amplification etc. In my opinion, another interpretation ...
1
vote
1answer
61 views

Linear Algebra Help! Matrices with respect to given basis

Hey Guys I'm new here so I dont know much exactly how to do all the fancy symbols but here it goes: Let $V_n$ be the vector space of real polynomials of degree at most $n$ and Let $B_n$ be the usual ...
0
votes
0answers
38 views

Rank of a large matrix

Suppose i want to calculate rank of a large $N\times N$ matrix having only $0$ and $1$ which is represented by $M$ segments each segment here is depicting that for a segment [LEFT,RIGHT] the row of ...
0
votes
3answers
46 views

Linear Algebra: Matrix Spanning/Consistency Question

1) If there are $5$ vectors found in $\mathbb{R}^7$ will these vectors Span $\mathbb{R}^7$? Please explain. 2) Give an example of a $3$ by $5$ matrix for which all systems, $Ax=b$ for any $b$ in ...
2
votes
1answer
48 views

How much linearly independent? or linearly dependent?

I want to improve a rank-deficient matrix by augmenting a row vector to it. However, unfortunately, I have only very 'similar' vectors.. For example, my matrix is somewhat like.. [ 1 1; 1 0.99] and ...
1
vote
2answers
172 views

linear independence and reduced row echelon form

If I can write vectors $(2,0,0)$ ,$ (1,-1,0)$ and $(0,1,1) $ as $\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1 \end{bmatrix}$ using reduced row echelon form does this means that they ...
2
votes
7answers
109 views

Does $\{u_1, u_2, u_3, u_4\}$ spanning $\mathbb R^3$ mean that $\{u_1,u_2,u_3\}$ also does? Since it's a subset?

Does $\{u_1, u_2, u_3, u_4\}$ spanning $\mathbb R^3$ mean that $\{u_1,u_2,u_3\}$ also does? Since it's a subset? A little unclear about this...
0
votes
3answers
336 views

Distance from Point on Plane to Origin

I have a practice question that is asking: Find the point on the plane 2x - 3y + z = 3 closest to the origin. What is the distance from that point to the origin? Here is my work so far (if you can ...
2
votes
1answer
40 views

Geometrically Describing a Subspace

I have a practice question here and it is asking to geometrically describe this subspace in $R^3$ It is asking if it is a point, a line, or a plane or all of three-space. Here it is: Span {(1,-2, 1) ...
0
votes
1answer
49 views

A little help on geometric description of $\Bbb R^2$ in linear algebra

I just started studying vectors in linear algebra, and I didn't understand the idea of the geometric description of a vector. Why do we treat the vector entries as coordinates? As far as I ...
0
votes
0answers
18 views

Difference between two “semi”projection.

Given $m<n$. Suppose that $H$ be a full rank $m \times n$ matrix. Put any basis of the subspace $S=\{x:Hx=0\}$ namely $\{K_1,\dots,K_{n-m}\}$ and define a matrix $K$ by $K=[K_1,\dots,K_{n-m}]$, ...
3
votes
3answers
70 views

Vector multiplication. Difference between scaler and dot product?

We just started a new class where the first topic is briefly talking about vectors and vector multiplication. All tying this into neural networks. I am a bit behind with the understanding of what the ...
0
votes
1answer
41 views

Finding the transformation matrix of this linear map.

I've being doing several exercises and none was of this kind, which I can't figure out: Let $V$ and $W$ be vector spaces with basis $B=\{\vec{v_1},\vec{v_2},\vec{v_3}\}$ and ...
0
votes
3answers
73 views

Why do vectors often have 1 dimenision as 1

In textbooks and school, why is it that vectors are often shown to be n x 1 or 1 x n matrices? I have rarely seen other dimensioned matrices representing matrices. \begin{bmatrix} 1 & 2 ...
1
vote
1answer
81 views

$Ax=v$, $Ax=w$ consistent; what about $Ax=v+w$?

Let A = \begin{bmatrix} 7 & -3 & 5\\ -4 & 1 & -5 \\ -5 &2 &-4 \end{bmatrix} , v = \begin{bmatrix} 2 &\\ 1\\ -1 \end{bmatrix}, and w = ...
1
vote
2answers
38 views

Get the parametric equation of a line

How I can get the parametric equation of this line? $$r\equiv\begin{cases} x-y=2 \\2x-z+1=0\end{cases}$$ Thanks.
0
votes
0answers
22 views

Geometric nature and Cartesian equation of a vector space

Determine the geometric nature and the Cartesian equation of the sub-vector space of $\Bbb R^2$ spanned by the first two column vector of A. A is a matrix defined by: $$A =\left( ...
2
votes
2answers
41 views

transforming a vector from cartesian to spherical and cylindrical co-ordinate system

I know the formula(which i don't know how to copy here but it was in matrix form) for transforming a vector from cartesian system to spherical or cylindrical coordinate system. But, I want to know its ...
0
votes
0answers
26 views

find the column space of a matrix

For my understanding, given a reduced matrix, the set of columns which has a pivot (leading coefficient) is spanning the column space. Is that correct? Can I count on it?