1
vote
1answer
31 views

Find Determinant of A

I've tried creating a triangular matrix, tried row reducing but can't figure it out as I keep on having c-unknown in my answer. How would I do this?
0
votes
1answer
21 views

Bilinear forms and scalar product

Forgive me for these may not be the correct English mathematical terms. Let $(V, \langle, \rangle)$ be a euclidean vector space of finite dimension $n$ and $f:V \times V \rightarrow \mathbb{R}$ a ...
-1
votes
0answers
10 views

Hybrid encryption RSA with AES?

A common variant of textbook RSA is the following: During key generation, the modulus N is chosen as usual. We chose e as e := 3 (instead of random). Then d is chosen with ed ≡ 1 mod φ ( N ) (as ...
0
votes
0answers
24 views

Why is $\hat{x}$ in the linear regression equation $A^TA\hat{x} = A^Tb$ part of $C(A^T)$

When finding the best fit line for a number of points, we use $A^TA\hat{x} = A^Tb$ where we solve for $\hat{x}$. I understand that the projection $p=A\hat{x}$ is part of the column-space of $A$ and ...
0
votes
1answer
29 views

question about span and basis

I have a question from homework (I'm not sure if my solution is correct): Let $V$ be a vector space over a field $\mathbb{F}$ and let $W$ be subspace of $V$. Let $u$ and $v$ be vectors in ...
1
vote
3answers
35 views

Standard matrix A of T?

Help please. What would be the standard matrix of A? I know how to do number 2 and 3 but I'm just having trouble with A. I asked this earlier but I lost my account and I'm not sure if I posted ...
1
vote
1answer
46 views

Prove that this matrix is not diagonalizable WITHOUT determinants

I have this matrix: $ \left( \begin{array}{cccc} 22 & 23 & 10 & -98\\ 12 & 18 & 16 & -38\\ -15 & -19 & -13 & 58 \\ 6 & 7 & 4 & -25 \end{array} \right) ...
-1
votes
2answers
38 views

Prove that the vectors $v_1,v_2,\ldots,v_k \operatorname{span}R^n$ if and only if $[v_1]_B,[v_2]_B,\ldots,[v_k]_B \operatorname{span}R^n$.

From section on Change of Basis $\longrightarrow$ Assume the vectors $v_1,v_2,\ldots,v_k\operatorname{span}R^n$, we must show that $[v_1]_B,[v_2]_B,\ldots,[v_k]_B\operatorname{span}R^n$. We can ...
0
votes
1answer
28 views

Determine which of the following subsets of $\Bbb{R}^n$ are subspaces of $\Bbb{R}^n (n>2)$.

I'm having a bit of trouble showing that the following subsets of $\Bbb{R}^n$ are subspaces of $\Bbb{R}^n (n>2)$. I know that I need to show that they are closed under addition and multiplication, ...
0
votes
2answers
41 views

Why does this form a basis for $V$? (Intuitive explanations please)

Let $V$ be the space spanned by $\mathbf f_1=\sin(x)$ and $\mathbf f_2=\cos(x)$. Show that $\mathbf g_1=2\sin(x)+\cos(x$) and $\mathbf g_2=3\cos(x)$ form a basis for $V$. We can see that $$\mathbf ...
0
votes
2answers
25 views

Show that the set $W$ of all polynomials in $P_2$ such that $p(1)=0$ is a subspace of $P_2$. Find a basis for $W$.

a.) Show that the set $W$ of all polynomials in $P_2$ such that $p(1)=0$ is a subspace of $P_2$. b.) Make a conjecture about the dimension of $W$. c.) Confirm your conjecture by finding a ...
0
votes
0answers
21 views

Linear algebra.Proof proportinal between minors and cofactors

$B$ is square matrix. Order of matrix $B$ is $n$. First $m$ lines form the matrix $C$, $rank (C)=m$.Last $n-m$ lines form fundamental system solutions of homogeneous linear equation with matrix $C$ ...
1
vote
3answers
66 views

An infinite generating set of a finite dimensional vector space contains a basis

Let $S$ be an infinite generating set of a finite dimensional vector space , then how do we prove that there is a subset of $S$ which is a basis of the vector space ? Please help
0
votes
1answer
26 views

Show that the entries of a matrix are:

For a regression model $y=\beta x$ (note there is no intercept term), show that entries of the matrix $\bf{H} = \bf{X}[\bf{X'}\bf{X}]^{-1}\bf{X'}$ are $h_{ij} = ...
2
votes
1answer
54 views

Eigenvalues of the vectors of

I came across the following problem: "Let $\mathbf a\in\mathbf{R}^n$ be a fixed $n$-component real, non-zero, vector. Let $A^+$ and $A^−$ be real $n\times n$ matrices with components: $$(A^\pm)_{ij} ...
2
votes
2answers
59 views

Dual Vector Space embedding

Is there an embedding of any vector space $V$ into $V^*$? As far as I know it is not true. The statement that I know of is that there is natural embedding of $V$ into $V^{**}$ Is there any ...
0
votes
0answers
19 views

linear algebra - fourier coefficients of piecewise

Find fourier coefficients of given function: f(t) = {-1 if t $\leq$ 0; 1 if t > 0} so do I do this? $a_{0} = \int_{a+-\pi}^{a+\pi}1$, $a_{k} = \int_{a+-\pi}^{a+\pi}1*cos(kx)$, $b_{k} = ...
0
votes
3answers
50 views

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation. Find $T(x)$

Let $T : \mathbb{R}^3 \to \mathbb{R}^3$ be a linear trasformation with $T \left(\begin{bmatrix} 1 \\ -2 \\ -1 \\ \end{bmatrix}\right) = \begin{bmatrix} 1 \\ -1 \\ 2 \\ ...
0
votes
1answer
21 views

Prove that Orthogonal Set Is Linearly Independent

Suppose that $V$ is an inner-product space; $(\space ,\space )$ is our inner-product. I have seen many proofs that go as follows: Let $\{x_1, x_2 ,\ldots, x_n\}$ be orthogonal. Set $a_1x_1 + a_2x_2 ...
0
votes
1answer
23 views

$A$ is an $n \times n$ invertible matrix, prove that $f(\mathbf u, \mathbf v)= \mathbf u^TAA^T \mathbf v$ defines an inner product on $\mathbb R^n$

I have difficulty especially proving that $f(\mathbf v, \mathbf v) \geq 0$ for all $\mathbf v$. Thanks
0
votes
1answer
19 views

Show a set is in the null space of the transpose of A

I'm trying to show that for $A \in F^{MxN}$ (a matrix with the nth column $a_n$) the following set is in the null space of $A^T$, that is: $N(A^T) = \{x \in \Re^M : A^Tx = 0\} = \{x \in F^M : ...
2
votes
1answer
46 views

How does negating a matrix affect its eigenvalues?

I'm working on the following problem: "If $Ax = \lambda x$, find an eigenvalue and an eigenvector of $e^{At}$ and also of $-e^{-At}$." So far, I have figured that $e^{\lambda t}$ will be an ...
0
votes
1answer
33 views

Find values so matrix not invertible?

$$ \begin{pmatrix} 2 & 4 & k \\ 1 & 3 & 2 \\ 3 & k & 9 \\ \end{pmatrix} $$ For what values of $k$ is the above matrix not invertible. Need help. Don't know where to ...
0
votes
2answers
18 views

Linear Algebra: show $\sum_{m=1}^{M} a_m x_m = 0$ is a subspace

I have a problem that I can't get my head around. It says that a is any vector in $\mathbb{F}^M$ and to verify (by the three properties of subspaces) that $\sum_{m=1}^{M} a_{m}x_{m} =0$ is a subspace ...
2
votes
2answers
57 views

Prove that if $C=A+iB$ is invertible, then so is $A+\lambda B$ for some $\lambda$

I've got a homework question that I've honestly no idea how to tackle. It goes as: Let $A$, $B$ be real $n × n$ matrices such that the complex matrix $C = A + iB$ is invertible. By considering ...
1
vote
3answers
69 views

Solution for $x$ with exponents?

I am trying to solve the following, $$7^{(2x+1)} + (2(3)^x) - 56 = 0$$ Should I put the 56 on the other side and get the log of both sides and is there a better way to solve this.
0
votes
3answers
31 views

Choose h and k such that the system has a solution, a unique solution and many solutions.

Im learning linear algebra, and im tasked with choosing $h$ and $k$ such that this system: $$ \begin{cases} x_1+hx_2=2\\ 4x_1+8x_2=k\\ \end{cases} $$ Has (a) no solution, (b) a unique solution, and ...
0
votes
4answers
24 views

Find a basis for the subspace of $\Bbb{R}^3$ that is spanned by the vectors

Find a basis for the subspace of $\Bbb{R}^3$ that is spanned by the vectors: $$v_1=(1,0,0), \space v_2=(1,0,1), \space v_3=(2,0,1), \space v_4=(0,0,-1)$$ I am not sure how to solve this problem. I ...
0
votes
1answer
15 views

Interpretation of the basis and coordinates for this solution space.

I found $x_1=x_3$ and $x_2=0$. So, $x_1=t;x_2=0; x_3=t$ Therefore: $$(x_1,x_2,x_3)=t(1,0,1)$$ So, the dimension is $1$ and the basis is $(1,0,1)$. Now, I am having trouble interpreting this ...
0
votes
1answer
17 views

Dimension of solution space has 3 vectors but 6 components?

I am not understanding how this has dimension $3$, but there are six components in each vector. If $3$ vectors span the space, why are there more than $3$ components in each vector? I thought for a ...
0
votes
2answers
36 views

Proving That a $1 \times 1$ Matrix has a Rank of $1$

I have an assignment that asks me to prove something but I've hit a roadblock. Let $u$ be a $3 \times 1$ matrix with $u_{1}$, $u_{2}$, and $u_{3}$. Let $v$ be a $3 \times 1$ matrix with $v_{1}, ...
0
votes
1answer
34 views

Distance of a point to a plane

Let $T$ be the plane $x+2y+3z=11$. Find the shortest distance $d$ from the point $P=(2, 4, 5)$ to $T$, and the point $Q$ in $T$ that is closest to $P$. This is just one of the questions on my ...
2
votes
2answers
48 views

Is it possible to triangularize a matrix only by adding scalar multiples of rows to each other?

I am working on showing if $B$ is a $s \times s$ matrix, $D$ is a $t \times t$ matrix, $C$ is a $s \times t$ matrix, and $0$ is a $t \times s$ zero matrix, then $\det(A)=\det(B)\det(D)$, where $$A = ...
1
vote
1answer
24 views

vector projection on to subspaces

If given a subspace of R4 that is spanned by the set of orthogonal vectors W =span { (0,1,1,1),(1,1,0,-1) }. How to find the projection of a vector u onto the subspace? if u = (2,1,2,0)? What I have ...
1
vote
3answers
52 views

Finding the limit of a sequence by diagonalising a matrix

Consider the sequence described by: $\frac11 , \frac32 , \frac75 , ... ,\frac {a_{n}}{b_{n}}$ where $ a_{n+1} = a_n +2b_n $ and $b_{n+1} = a_n+b_n$ Find a matrix $A$ such that ...
2
votes
3answers
44 views

Evaluate determinant of an $n \times n$-Matrix

I have the following task: Let $K$ be a field, $n \in \mathbb{N}$ and $a,b \in K^n$. Evaluate the determinant of the following matrix: $$\begin{pmatrix} a_1+b_1 & b_2 & b_3 & \dots ...
2
votes
0answers
34 views

Show $\langle , \rangle |_W$ non degenerate $\implies$ $\langle , \rangle |_{W^\perp}$ non degenerate

Let $W \subset V$ be a subspace and $\dim V < \infty$. If $\langle , \rangle$ and the restriction $\langle , \rangle |_W$ are non degenerate, then $\langle , \rangle |_{W^\perp}$ is non degenerate ...
2
votes
2answers
52 views

Systems of Linear Differential Equations - Is this Correct?

I have to solve the following first-order linear system, $x(t)$ represents one population and the $y(t)$ represents another population that lives in the same ecosystem: (Note: $'$ denotes prime) ...
0
votes
1answer
24 views

Linear map over a vector space of polynomials

Let $F$ be a field and Let $F_{n+1} [X]$ (odd notation, in my opinion) be the vector space of polynomials of degree less than or equal to $n$ over $F$. Define $t: F_{n+1}[X] \to F_{n+1}[X]$ by ...
3
votes
1answer
41 views

Systems of Linear Differential Equations - population models

I have to solve the following first-order linear system, $x(t)$ represents one population and the $y(t)$ represents another population that lives in the same ecosystem: (Note: $'$ denotes prime) ...
1
vote
1answer
28 views

What are the Routh Hurwtiz Criteria for 3$\times$3 Matrices?

The Criteria I know (for dynamical systems) is... The eigenvalues of a matrix are guaranteed to be negative if Tr($J$)<0 and det($J$)>0, where $J$ is the Jacobian of some 2 dimensional dynamical ...
0
votes
0answers
26 views

Parity check matrix operations

Let $C_1$ and $C_2$ be linear codes of the same length over the finite field $F$, and let $H_1$ and $H_2$ be parity-check matrices of $C_1$ and $C_2$ respectively. Define $C_3$ as the code $C_3 = ...
2
votes
2answers
37 views

Is $u^TAu \geq 0$ true for all symmetric matrices $A$?

we know from the definition of inner product and norm, that $u^Tu$ is always larger than zero, except the case where $u=0$ at which case it is zero. I came across a question that infers that $u^TAu ...
0
votes
3answers
39 views

Calculate Matrix A from eigenvalues, but no given eigenvectors

Here is my question: Write down a nontriangular 3 by 3 matrix whose eigenvalues are 6, 9, 2. I understand that you can calulate Matrix A using the formula A=V$\Lambda$$V^-1$, but is there a way to ...
2
votes
2answers
43 views

For a shift matrix $A$, prove that $A^n=0$ but $A^{n-1} \neq 0$.

Let $A\in F_n$ be the matrix $\begin{pmatrix} 0&1&0&0&\cdots&0 \\ 0&0&1&0&\cdots&0 \\ \vdots\\ 0&0&0&0&\cdots&0 \end{pmatrix}$, whose ...
2
votes
2answers
33 views

Definition: Eigenvalues of a matrix

1) Can a non-square matrix have eigenvalues? Why? 2) True or false: If the characteristic polynomial of a matrix A is p($\lambda$)=$\lambda$^2+1, then A is invertible. Thank you!
1
vote
1answer
18 views

Trouble understanding finite vector spaces and Gaussian coefficent

I have studied linear algebra for 2 months now and i cannot understand a task that i am currently trying to solve. Basically i am trying to find the amount of bases for n-dimensional vector space over ...
0
votes
1answer
14 views

Matrix Transformation - Using matrix multiplication

How do I use matrix multiplication to find the reflection of (-1,2) about the x axis, y axis and the line y=x?
0
votes
1answer
40 views

Finding basis of vector spaces

Without proof find the dimension and a basis of the following vector spaces $V$ over the given field $K$. $V$ is the set of all polynomials over $\mathbb{R}$ of degree at most $n$, in which the sum of ...
0
votes
3answers
54 views

For which values of $k$, we have $A = A^{-1}$?

I got this question in hw. Can anyone help me solve it? Let $ A = \left( \begin{array}{ccc} k & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & k \end{array} \right) $ For which values of ...