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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Linear Algebra. Bonus question Final Exam

This is from a practice final exam. I was wondering about the the 2nd part of A). It states that when v can't = O, it will form a basis. I'm having a tough time understanding that. How is that ...
2
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2answers
118 views

On the determinant of a certain matrix over the polynomial ring of $n$ variables over a field

Let $A = k[x_1,\dots, x_n]$ be a polynomial ring over a field $k$. Let $\sigma_1,\dots,\sigma_n$ be distinct permutations of the set $\{1,\dots,n\}$. Is the determinant det$(x_{\sigma_i(j)})$ ...
4
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2answers
95 views

Finding a basis for subspace of polynomials

Let $V=\mathscr{P}_{3}$ be the vector space of polynomials of degree 3. Let W be the subspace of polynomials p(x) such that p(0)= 0 and p(1)= 0. Find a basis for W. Extend the basis to a basis of V. ...
1
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1answer
21 views

Find an orthogonal matrix such that its first line is $\frac{1}{5},\frac{2}{5}$

An orthogonal matrix is one matrix $A$ such that $A^t = A^{-1}$. So what I did: Suppose: $$A = \begin{bmatrix}\frac{1}{5}&\frac{2}{5}\\x&y\end{bmatrix}$$ Then: ...
3
votes
1answer
87 views

Linear algebra gram-schmidt

Will the reordering of the original basis before starting the Gram-Schmidt process lead to the same orthogonal basis? Is there an obvious proof for this one or is this clear already?
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71 views

Spectral radius of matrix from SOR method

Suppose we write a matrix $A = L + D + U$ with lower triangular, diagonal and upper triangular parts. When trying to solve the equation $Ax=b$, we use a successive overrelaxation technique such that ...
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2answers
63 views

Frobenius Norm to L2 norm Problem

Here is the problem: if $v^1$, $v^2$, ..., $v^d$ is an orthonormal basis in $\mathbb{R}^d$, then show that $$ ||A - A\sum_{i = 1}^k v^i(v^i)^T ||^2_F = \sum_{i = k+1}^d||Av^i ||_2^2. $$ I am having ...
3
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29 views

What is the Coefficient Matrix of $T(p(t))=\int_0^t\int_0^yp(x)dxdy$ that maps $P_3\rightarrow P_5$?

The usual basis for $P_n$, of course, is given by $\left\{1,t,t^2,\cdots,t^n\right\}$. Why is the integrand a function of $x$? Does this matter for the purposes of constructing a change of basis ...
0
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1answer
45 views

Among all unit vectors $\vec{u}=\scriptsize\begin{bmatrix}x\\y\\z\end{bmatrix} \in \mathbb{R}^3$ which one minimizes the sum $x+2y+3z$?

My first instinct, of course, was to use Lagrange multipliers, but I have to use linear algebra to solve this. How would I construct an orthonormal basis in this case? I'm not sure how to ...
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2answers
70 views

What am I doing wrong in attempting to find the least squares solution of the system Ax = b?

I am attempting to find the least-squares solution x* of the system Ax = b, where $$ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \\ ...
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1answer
21 views

A website that determines whether there is a solution to a set of equations

Does anyone know a website that determines whether there is a solution to a set of equations? For calculus I like to use derivative calculators to see whether my answers are correct and I would to do ...
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2answers
40 views

Linear Tranformations

Hello, everyone. I'm having trouble solving this problem. I tried row reducing and find that the 3rd row can be reduced to only zeroes. Which leaves two pivots and two free variables, but the ...
0
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0answers
43 views

Calculating covariance (PCA example)

What is $z_1$? We have $n$ observations of $p$ different quantities (variables). It's defined on page 6. Is it just a projection onto the vector $a_1$? $a_1$ is our new basis vector and thus $z_1$ is ...
0
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1answer
33 views

Changing the subject of a formula involving the floor.

I'm trying to prove if $3\left\lfloor \frac{x+1}{2}\right\rfloor$ is onto. But I cannot seem to be able to change the subject of my formula to $x$ from $y=3\left\lfloor \frac{x+1}{2}\right\rfloor$. I ...
5
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0answers
105 views

Lie derivative and simultaneous diagonalizability

I just arrived at this theorem: Let $M$ be an $n$-manifold and let $\{X_j\}_{j\le k}$ be a collecion of $k$ vector fields and $p \in V \subset M$ satisfying: 1) $\{X_j(p)\}_{j\le k}$ is ...
0
votes
1answer
50 views

Extension of an Isometry

I'm really stucked in this problem Let $\mathbb{S}^2=\{(x,y,z)\mid x^2+y^2+z^2=1\}$ the unit sphere in $\mathbb{R}^3$, and let $A:\mathbb{S}^2\to\mathbb{S}^2$ an Isometry. Show that $A$ can be ...
0
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3answers
60 views

Understanding being closed under addition and multiplication

I am having serious trouble trying to figure out this whole, "being closed under addition and scalar multiplication" Our example is Let $W=\{[a,b,c]:a+b=4c;b=2c\}$ I need to figure out if it is ...
1
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1answer
80 views

Solving parametric linear equation

I am trying to solve the following system of equations using linear algebra: $$ x -2y +4z = a\\ 5x -y +z = b\\ -x +3y -z = w\\ $$ So how can I tell if the equation has infinite solutions? ...
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2answers
40 views

Determinant of a Matrix given the Characteristic Equation

If the characteristic polynomial of a matrix A is... p(λ)=(λ+1) (λ−2)^2 (λ+3)^2 Find det(A^−1). Thanks!
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1answer
31 views

showing equality of dimensions

Let $\alpha \in \mathbb{C}$ be a complex number. Let $V = \mathbb{Q}(\alpha)$ be the rational vector space spanned by powers of $\alpha$. That is $V = <1,\alpha,\alpha^2,\ldots>$. If $P(t)$ ...
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1answer
42 views

Prove one situation happens

Let $f\in End(V)$ of rank 2 where $V$ is over $\mathbb{C}^n$ field. Prove one of situation happens: 1) $f$ is diagonalised 2 $tr(f)$ is an eigenvalue of f 3) $\frac{1}{2} tr(f)$ is an eigenvalue of ...
0
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2answers
43 views

Homework Problem About Finding a Value of $k$ for Which the Given System of Equations Has No Solutions

While working through the third of three packets I'm going through to review for a pre-test for an independent-study calculus class, I came across the following problem: For what value of $k$ ...
0
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1answer
37 views

Is it a vector or scalar?

Let $X=(x_1,x_2,...,x_p)$ and $a_1=(a_{11}, a_{21},...,a_{p1})$. What is $z_1=a_{1}^TX$? Obviously it's a square matrix. However, in the text I'm reading it says ...
0
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2answers
51 views

2×2 matrix with one single eigenvalue, but two linearly independent eigenvectors?

Can we have a 2×2 matrix with one single eigenvalue, but two linearly independent eigenvectors? Is this possible? If so, how? Thanks for the help!
0
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4answers
65 views

Trouble understanding Linear transformation question

I am having some trouble figuring out how to solve a question It is , consider a linear transformation $T: P_2(t) \to P_2(t)$ given by $T(p(t))=(t-1)\frac{dP}{dt}(t)$ What can we say about this, ...
0
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1answer
31 views

Is it defined the product of vectors of different spaces?

I know that the sum of vectors of different spaces is not defined, but what about the multiplication of vectors of different spaces. For example, what about the multiplication of $v_1 = ...
0
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1answer
22 views

Does $\vec{b}$ have to be in $\text{im }(A)$ if $|| \vec{b}-A\vec{x}^*||=0$?

I'm going through a least squares computation where $A=\begin{bmatrix}3&2\\5&3\\4&5\end{bmatrix}$ and $\vec{b}=\begin{bmatrix}5\\9\\2\end{bmatrix}$. From ...
1
vote
1answer
84 views

Find a isometry such that the matrix in respect to the canonical basis is:

I need to find a isometry such that the matrix in respect to the canonical basis is: $$\begin{bmatrix}\frac{1}{\sqrt{2}}& \frac{1}{\sqrt{2}}& 0\\0 & 0 & 1\\x & y & ...
2
votes
1answer
249 views

show invariant subspace is direct sum decomposition

Let $f \in End(V)$ ($V$ is a finite dim.) be diagonalized where $a_1, … ,a_k$ are eigenvalues and for $i \neq j$ we have $a_i \neq a_j$. Prove for every subspace $f$ invariant $W \subset V$ holds ...
0
votes
1answer
126 views

For real matrices, if $A$ and $B$ are both positive-definite, show that all of $AB$'s eigenvalues are positive.

The original question goes equivalently like this For real matrices, if $A$ and $B$ are both positive-definite Prove: all the eigenvalues of $AB$ are positive. Facts that I know may have a ...
0
votes
1answer
73 views

Find Cartesian and Vector equations of the plane perpendicular to (1, 0, -2) and containing the point (1, -1, -3)

I'll work through my current progress until I reach the bit where I get stuck. We are given $$n = (1, 0, -2)$$ Thus Cartesian form will be $$x - 2z = r\cdot n$$ Now, $$r\cdot n = (1, -1, -3)\cdot ...
2
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1answer
32 views

Inequality $|A+B|_m\leq|A|_m+|B|_m$ on square matrices

Consider $n\times n$ real matrices $A$ and $B$. If $|A|_m$ denotes the modulus matrix of $A=[a_{i,j}]_{n\times n}$, and is defined as $|A|_m := [|a_{i,j}|]_{n\times n}$, prove that ...
1
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0answers
51 views

proving Pythagoras Theorem in the third dimension using orthogonal projection from a parallelogram

how can i prove $m^2 + n^2 + o^2 = p^2$ ? given $u = (u1, u2, u3)$ and $v = (v1, v2, v3)$ be vectors that span the parallelogram P $m$ represents the area of the orthogonal projection in the ...
1
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1answer
44 views

How do I prove that $A^{t}A$ is not invertible?

Suppose A is an $n × m$ matrix. Prove that if $n < m$ then $A^{T} A$ is not invertible.
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2answers
30 views

Basis for $\mathbb R^n$ and invertible $n\times n$ matrix

Let $s = \{v_1, v_2, v_3, ... , v_n\}$ be a basis for $\mathbb R^n$. Suppose $B$ is an invertible $n\times n$ matrix, show that $w = \{ Bv_1, Bv_2, Bv_3, ...,Bv_n\}$ is also a basis. I know that you ...
0
votes
1answer
55 views

Complete the following proof that $-u$ is the unique vector in V such that $u+(-u)=0$.

suppose that $w$ satisfies $u+w=0$. Adding $-u$ to both sides we have $(-u)+[u+w]=(-u)+0$ $[(-u)+u]+w=(-u)+0$ $0+w=(-u)+0$ $w=-u$
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votes
3answers
130 views

If $x^2 +px +1$ is a factor of $ ax^3 +bx+c$ then relate $a,b,c$

Suppose If $x^2 +px +1$ is a factor of $ax^3 +bx+c$ then relate $a,b,c$ such that $a,b,c \in R$ I can write $$ax^3 +bx+c=(x^2 +px +1)(\lambda x +D)$$ $$\implies ax^3 +bx+c =\lambda x^3 + x^2.p\lambda ...
1
vote
1answer
394 views

Is the formula $(\text{ker }A)^\perp=\text{im }A^T$ necessarily true?

If $A$ is a $n\times m$ matrix, is the formula $(\text{ker }A)^\perp=\text{im }A^T$ necessarily true? I'm thinking that rank-nullity would be the simplest and easiest way to prove this, but would ...
4
votes
2answers
41 views

Euclidean norm of complex vectors

I am working on a proof: One has two vectors, $u,v \in \mathbb C^n$, such that $u \cdot v=0$ . I am trying to prove that $$|u + v|^2 = |u|^2 + |v|^2.$$ I am a little stuck on how to do $u + v$ ...
0
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2answers
31 views

Factor nonsingular symmetrice square matrix $M$ into $A^TA$

I have a matrix $M$ of size $n,n$ consisting of real numbers. It is nonsingular and symmetrice, i.e. $M = M^T$. Does there exist a factorization whereby $M = A^TA$? I know I can't use Cholesky ...
0
votes
1answer
48 views

Rank of an element of the exterior power

Let $X$ be a finite-dimensional vector space and let $\Lambda^p(X)$ be the $p$th exterior power of $X$. My picture of an elementary element $x_1 \wedge \ldots \wedge x_p$ in the exterior power is ...
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1answer
67 views

Line plane intersection

I have two planes in $\mathbb R^3$ as shown below: axes representation corrected after MvG's comment Each plane is a finite area, a rectangle with length and width $H_l, H_w$. Each plane has its ...
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1answer
57 views

Under what conditions on a, b, c in R is the following set A linearly dependent?

These are 2x2 matrices. I have the answer to the question above but Im not exactly sure how that answer was gotten, if someone could explain it, it would be quite helpful. Thanks in advance. ...
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1answer
98 views

operation to get a diagonal matrix from a vector

In many programs you can create diagonal matrix from a vector, like diag function in Matlab and DiagonalMatrix function in Mathematica. I'm wondering whether we can use matrix product (or hadamard ...
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0answers
88 views

Why is the border rank and rank different for order 3 tensors and above?

Recall the definition of border-rank of a tensor T: border-rank(T) = the minimum r such that $\forall \epsilon > 0$ there exists an approximate tensor $T' = \sum^r_{i=1} u_i \otimes v_i \otimes ...
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2answers
33 views

How to extract transition matrices for smaller intervals than the original transition matrix?

I have a transition matrix, for example: \begin{bmatrix} 0.95 & 0.03 & 0.02 & 0 &0 \\ 0 &0.90 &0.1 &0 & 0\\ 0 &0.05 & 0.80 &0.1 &0.05 \\ 0& ...
0
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0answers
18 views

Can $\mathbb E[\theta|x]=(a+2n)/(b+n\bar x)$ be written as $\mathbb E[\theta|x]=t\hat \theta + (1-t)\mathbb E[\theta]$

Can $\mathbb E[\theta|x]=\frac{a+2n}{b+n\bar x}$ be written as $\mathbb E[\theta|x]=t\hat \theta + (1-t)\mathbb E[\theta]$, when $\hat\theta=\frac{2}{\bar x}$ and $\mathbb E[\theta]= \frac{a}{b}$? I ...
0
votes
1answer
63 views

Finding length of side on parallelogram

A parallelogram has sides $AB$, $BC$, $CD$, and $DA$. Given $A(1,-1,2)$, $C(2,1,0)$, and the midpoint $M(2,0,-3)$ of $AB$. Find $BD$. I am unsure how to solve this question with the given midpoint ...
0
votes
1answer
95 views

Differential of an operator $\phi: Mat_{2 \times 2}{\mathbb{R}} \rightarrow Mat_{2 \times 2}{\mathbb{R}}$

Let's consider an operator $ \phi: Mat_{2 \times 2}{\mathbb{R}} \rightarrow Mat_{2 \times 2}{\mathbb{R}}$ so that $A \rightarrow A^{-1}$. How to evaluate its differential? By the differential we ...