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 ...

learn more… | top users | synonyms (1)

0
votes
1answer
22 views

Given two column vectors $a$ and $b$, what is the determinant of $A$ if $A=Id-ab^T$

Given two column vectors $a$ and $b$ in $\mathbb R^n$ , $n \ge 2$, form the $n×n$ matrix and $I_n$ the identity matrix. Let be $A = I_n-ab^T$. What is the determinant of $A$?
1
vote
1answer
21 views

The sum of $V=U+W$ of a vectorspace V and subspaces $U$, and $V$

I know what the sum of two subspaces is and how we notate but is it ok to write a minus to denote what I hope should be obvious is meant. So we have $V=U+W+Y$ where $V$ is a v.space and $U,W,Y$ ...
0
votes
1answer
26 views

Let V be an inner product space.Then for x,y,z belongs to V and belongs to field,F,the following statements are true.

(a) $\langle x,y+z\rangle=\langle x,y\rangle+\langle x,z\rangle$. (b) $\langle x,cy\rangle =\bar c\langle x,y\rangle$. (c) $\langle x,0\rangle = \langle 0,x\rangle =0$. (d) $\langle x,x\rangle=0$ ...
1
vote
1answer
12 views

There is only one linear function whose image of a specific base is a specific set of vectors.

So I found this theorem: Let $V$ be a real vector space of dimension $n$, and $\mathscr{B} = \{\mathbf{b}_1, \dotsc, \mathbf{b}n\}$ its base; let $V'$ be a real vector space and $\mathbf{c}_1, ...
6
votes
1answer
73 views

$A^2+B^2=AB$ and $BA-AB$ is non-singular

The question is: Are there square matrices $A,B$ over $\mathbb{C}$ s.t. $A^2+B^2=AB$ and $BA-AB$ is non-singular? From $A^2+B^2=AB$ one could obtain $A^3+B^3=0$. Can we get something from this? ...
4
votes
2answers
117 views

Uniform unboundedness of linear operators

Question: Suppose that $(T_k)_{k=1}^{\infty}$ is a sequence of invertible linear operators on $\mathbb{R}^n$. Suppose that $\forall x \in \mathbb{R}^{n}\setminus \{0\}$, we have $$\lim_{k\to\infty} ...
1
vote
2answers
32 views

Problem in solving a question of vector space.

The question is : Let, $V$ be the subspace of all real $n \times n$ matrices such that the entries in every row add up to zero and the entries in every column also add up to zero. What is the ...
1
vote
1answer
31 views

Decomposing vector space into positive/negative definite subspaces

Consider the quadratic form: $$f:\mathbb{R}^3\to\mathbb{R};\quad (x,y,z)\mapsto x^2+2y^2-2xy-2xz$$ I am doing a problem which asks me to find subspaces $A,B\subseteq \mathbb{R}^3$ such that ...
3
votes
3answers
37 views

Show that $\dim(\operatorname{range}(T)) = 1$.

Let $T :\mathbb{R}^3 \to \mathbb{R}^3 $ be a linear transformation such that $T \neq 0$ but $T^2=0$. Show that $$\dim(\operatorname{range}(T))=1$$
0
votes
0answers
28 views

Derivative of $\frac{dA^TA}{dx}$ [duplicate]

I think I have an easy question but I can't understand how to do. I am trying to figure out how to do the derivative of a symmetric matrix $B = A^TA$ w.r.t. one parameter $x$ of the matrix: ...
1
vote
1answer
18 views

Spectral Theorem for a Complex Vector Space and corollary

When it says orthogonal projections it must mean they are orthogonal with each other otherwise if it meant they were orthogonal lin transformations then they would be invertible- the only invertible ...
0
votes
2answers
55 views

Intersection of 2D planes in 4D space

If I had a four dimensional space, in which I embedded two planes, what possible intersections could they have? Constructing a Plane To give this more context, consider the following. If I had a 4d ...
-1
votes
0answers
8 views

Symmetric Linear Transformation Matrix Multiplication

Assume that S is a matrix in $ \mathbb R^{nxn} $. Prove that $(S \vec x)^T S \vec y = \vec x \cdot \vec y $ I understand that if S is a symmetric matrix, then the transpose of S equals S. I don't ...
0
votes
1answer
13 views

SVD and homogeneous equation

Suppose a $m \times n$ matrix $A$, and column vector $h$. ($A$'s rank is equal or smaller then $n$(=$h$'s length).) If, $$ Ah=0 $$ then $h$ can be the last column of $V$ where $A = UDV^T $. ...
0
votes
1answer
43 views

Prove or disprove: If $1=||A||>||B||$, then $A-B$ is nonsingular.

Prove or disprove: If $1=\|A\|>\|B\|$, then $A-B$ is nonsingular. I think that since $\|A-B\|>0$ by the given conditions we know it is nonsingular. Any solutions or hints are greatly ...
1
vote
2answers
50 views

Does every invertible matrix A has a matrix B such that A=Adj(B)?

I'm trying to understand if it's always true, always true over $\mathbb C$ or never true. I know that if $A$ is invertible, than there exists $A^{-1}$. $$A=\frac{1}{det (A^{-1})}Adj(A^{-1})$$ So I ...
0
votes
1answer
12 views

Should the correlation PCA projection be computed on original or normalized samples?

Suppose we compute the correlation PCA of a dataset $X$ (with $m$ variables and $n$ observations) by first normalizing the input variables. That is: mean $\rightarrow 0$ and standard deviation ...
0
votes
0answers
19 views

Inverse problem of interpolation with Vandermonde matrix

Assume the following equation holds when $x, a, b,c,d$(variables) taking any values: $$ \begin{bmatrix} r_0&r_1&r_2 \\ s_0&s_1&s_2\\t_0&t_1&t_2 \end{bmatrix} \begin{bmatrix} ...
1
vote
0answers
32 views

Reconstruct matrix from adjoint

Is it possible to determine the entries of a matrix $A\in M_n(\mathbb R)$ knowing only its eigenvalues and some (but not all) entries of the adjoint $A^{\mbox{adj}}$? In particular how many entries of ...
0
votes
1answer
26 views

Constructing representation of $G$

Say we are given an arbitrary group $G$ and an arbitrary vector space $V$ over some field. How can we construct a representation of $G$ on some vector space from this data? Initially I wanted to ...
0
votes
1answer
11 views

matrices similarity and represantation matrix according to basis

Let $V$ be an $n$-dimensional vector space over the field $F$, and let $B = \{a_1 , a_2, ... , a_n \}$ be an ordered basis for $V$. According to the theorem there is the unique linear operator $T$ on ...
1
vote
2answers
25 views

Suppose the characteristic polynomial is $x^4$. Is it possible to get a jordan block of size$ J(2)J(2)$?

Suppose the characteristic polynomial is $x^4$. Is it possible to get a jordan block of size $ J(2)J(2)$? This would mean that I get $2$ vectors in the eigenspaces, but then it seems like an ...
0
votes
1answer
13 views

Clarification of Direct sum meaning with $\geq 3$ subspaces

Let $V$ be a vectorspace and $U_i$ subspaces of V. In the definition of $\oplus_{i \in I} U_i$ it is said that is does not suffice for $U_i$ to be pairwise disjoint. Instead we must have the stronger ...
-1
votes
0answers
24 views

conformal mapping of an ellipse to a circle

I am trying to map exterior of an ellipse to a circle using the transformation function $w=\frac{ze^m+\frac{e^m}{z}}{2}$. The major and minor axes of an ellipse is $a=cosh(m)$ and $b=sinh(m)$. I would ...
5
votes
0answers
40 views

Symmetric matrix and power of two

This is my first post. Given an square matrix $n\times n$, such that the elements on the main diagonal are 1, each row an each column has exact one time a number in $\{1,2,...,n\}$, the first column ...
1
vote
0answers
62 views

System of Equations which can be solved by inequalities: $(x^3+y^3)(y^3+z^3)(z^3+x^3)=8$, $\frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}=\frac32$.

S367. Solve in positive real numbers the system of equations: \begin{gather*} (x^3+y^3)(y^3+z^3)(z^3+x^3)=8,\\ \frac{x^2}{x+y}+\frac{y^2}{y+z}+\frac{z^2}{z+x}=\frac32. \end{gather*} Proposed by ...
0
votes
1answer
11 views

Formula for calculating markup with big % for small amounts and small % for larger amounts

I am trying to come up with a formula for calculating markup for products that range in value from a few cents up to tens of Dollars. At 10c I would like the markup to be around 500%, and from 2 ...
1
vote
2answers
15 views

Find the matrix of the given linear transformation $T$ with respect to a given basis.

How do you solve $T (f(t)) = f(2t - 1)$ from $P_2$ to $P_2$, with respect to basis $\beta = (1, t-1, (t-1)^2)$?
-3
votes
1answer
17 views

Find the matrix of the given linear transformation T with respect to the given basis. [on hold]

Question I'm not sure where to start with number 6. Can someone help? Thanks!
2
votes
1answer
22 views

Prove multidimensional Newton's method converge at least quadratically

Newton's method for root finding is simply $x_{n+1}=x_n-\frac{f(x_n)}{f'(x_n)}$. The following is a theorem from my textbook. where 6.1.22 is shown below Now I want to prove a similar ...
0
votes
3answers
45 views

find a vector perpendicular to set of vectors

I have a set of $m$ vectors given by $$A =\left(\vec{v}_0,\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_m\right),$$ elements of which need not be all linearly independent, in an $n$-dimensional space. I want a ...
1
vote
1answer
17 views

Prove that the inner product $\langle f,g\rangle= \int f(t)g(t) \, dt $ satisfies the property positive definiteness [duplicate]

In $C[a,b]$, define the product $\langle f,g\rangle= \int f(t)g(t) \, dt $. Show that this product satisfies the property, $\langle f,f\rangle$ is greater than zero for all non zero $f$ using a ...
2
votes
1answer
37 views

Prove that if $\|A\|<1$, then $\|(I-A)^{-1}\|\geq {1\over1+\|A\|}$.

Prove that if $\|A\|<1$, then $\|(I-A)^{-1}\|\geq {1\over1+\|A\|}$. I'm not sure how to prove this result. I see feel like a geometric series is involved though. Any solutions or hints are ...
1
vote
2answers
22 views

why any vector can be wriiten as the sum of two components in the row space and nullspace?

My textbook says that: there is a $m\times n$ matrix A, any vector x in $R^n$ can be written as the sum of a component $x_r$, in the row space, and a component $x_n$ in the nullspace: $$x=x_r+x_n$$ ...
1
vote
2answers
30 views

why nullspace is the largest subspace perpendicular to the row space?

The proof from my textbook is "If x were a vector orthogonal to the row space, but not in the nullspace, then the dimension of $C(A^T)^\perp$ would be at least n — r + 1. But this would be too large ...
0
votes
1answer
20 views

Find a vector in $\mathbb{R}^m$ that satisfies the following

Let $A$ be a full rank $m\times n$ matrix. a) Assuming that $m<n$, find a vector $x\in \mathbb{R}^n$ with smallest 2-norm that solves $Ax = b$. b) Assuming $m>n$, find a vector $ x \in ...
0
votes
0answers
4 views

Bounding Product of Nonnegative Matrices

Let $A \in \mathbb{R}^{n \times m}_{\geq0}$, not necessarily square, have nonnegative elements, and let $x \in \mathbb{R}^m_{\geq 0}$ be nonnegative. Clearly $Ax$ is a nonnegative vector. Suppose in ...
1
vote
1answer
21 views

Which of the following is a vector subspace of $R^3$

To prove that $F$ is a K-vector subspace of $ E$ it suffices to prove $\alpha f_1+\beta f_2 \in F$ with $(\alpha, \beta)$ $\in K²$ and $f_1 , f_2 \in F$ . For trivial cases and easy ones it seems ...
-5
votes
0answers
20 views

Subspace in R3 that has dimension 2 [on hold]

An example of a subspace in R^3 that has dimension 2 and contains the vector (1,0,1).
0
votes
0answers
12 views

How do I calculate $\vec{b_1},\vec{b_2},\vec{b_3}$? [on hold]

$U=\lambda ((1, 0, 1, 0)^T,(1, 1, 0, 1)^T,(1, -1, 1; 0)^T)$ Is a subspace of $\mathbb{R}^4$. Determine for $\vec{v} = (1, 1, 1, 1)^T$ the vector $ \vec{u}\in U $ with minimal $\left \| ...
0
votes
1answer
27 views

If A is invertible and orthogonally diagonalizable, is $A^{-1}$ orthogonally diagonalizable as well?

I know that the answer is yes. Are the reciprocal of the eigenvalues of A the eigenvalues for $A^{-1}$? If the eigenvalues for A are $3$ and $2$, would the eigenvalues for $A^{-1}$ be $1/3$ and $1/2$? ...
9
votes
2answers
121 views

Showing $A+B$ is invertible?

Question number two of this released exam asks: Let $A$, $B$ be two $n \times n$ matrices with real elements such that $A^3 = B^5 = I_n$ and $AB = BA$. Prove that $A+B$ is invertible. I am not ...
0
votes
0answers
12 views

Symmetric Positive Matrix Diagonal Value relationships after Gaussian Elimination

Let $A\in\mathbb{R}^{n\times n}$ be a symmetric positive definite matrix. Let Gaussian elimination be carried out on $A$ without pivoting. After $k$ steps, $A$ will be reduced to the form ...
0
votes
0answers
10 views

How do I calculate $\left \| \vec{v}-\vec{u} \right \|_2$ for $\vec{u}\in U$?

Let $U=\lambda ((1, 0, 1, 0)^T,(1, 1, 0, 1)^T,(1, -1, 1; 0)^T)$ Is a subspace of $\mathbb{R}^4$. Determine for $\vec{v} = (1, 1, 1, 1)^T$ the vector $ \vec{u}\in U $ with minimal $\left \| ...
1
vote
2answers
24 views

Determining similar matrices

I have this matrix $$A= \begin{bmatrix}1 &0& 2\\0&-1&-2\\2&-2&0\end{bmatrix}$$ I found the eigenvalues to be $0, 3, -3$ I am tasked with finding if $A$ is similar to a ...
0
votes
0answers
18 views

Simplification of vector cross product

If I have $-\mathbf{N}_u = a\mathbf{x_u} + b\mathbf{x_v}$ $-\mathbf{N}_v = c\mathbf{x_u} + d\mathbf{x_v}$ Then, why is $\mathbf{x_u} \times \mathbf{N_v} + \mathbf{N_u} \times \mathbf{x_v} = ...
0
votes
1answer
21 views

The ball of radius one with center at $(0,0,0,0,0)$ in $\mathbb{F}_{2}$ consists of $(0,0,0,0,0)$ and all the words weight one. For $w=(1,0,1,1,0)$,

definition : Let $w$ be a word in $\mathbb{F}_{q}^{n}$ and $r$ a natural number. The ball of radius $r$ with center $w$, denoted by $B_{r}(w)=\{x \in \mathbb{F}_{q}^{n} : d(w,x) \leq r\}$ . Now ...
-3
votes
0answers
26 views

Show that $B^T \cdot B =I$ [on hold]

Let $A\in \operatorname{Mat}_{m,n} (\mathbb R)$ be a real matrix of rank $r=n$. Let $(b_1,b_2,..,b_n)$ be an orthonormal basis for the column space $R(A)$ (terms. the scalar product) Let $C\in ...
0
votes
2answers
30 views

How do I show that a given lambda value is an eigen value of a matrix?

Also how do I show that a matrix is not diagonalizable based on my calculations from the equation $\det(A-\lambda I)=0$ For the first part say the matrix is: $$ \begin{pmatrix} 1 ...
0
votes
0answers
18 views

Need help understanding the Basis Theorem and applying it in computation

The Basis theorem is stated as follows: Let $V$ be a $p$-dimensional vector space where $p$ is equal to or greater than $1$. Any linearly independent set of exactly $P$ elements in $V$ is ...