1
vote
0answers
15 views

finding the symmetric point

let there be $4$ points. $A(-1,1,1), B(2,0,-1), C(1,3,-2), D(-2,-1,0)$. the $4$ points are not on the same line. the plane which goes through the points $A$ and $B$, and which is also paralel to the ...
1
vote
1answer
26 views

Find the triangular matrix and determinant.

I have a 4x4 matrix and I want to find the triangular matrix (lower half entries are zero). $$A= \begin{bmatrix} 2 & -8 & 6 & 8\\ 3 & -9 & 5 & 10\\ -3 & 0 & 1 & ...
0
votes
0answers
32 views

Composition of injective linear maps.

I was looking at some solutions for my homework and I didn't understand this part: $S_1,\ldots,S_n$ are injective linear maps such that $S_1S_2 \dots S_n$ makes sense. Prove $S_1S_2 \dots S_n$ is ...
1
vote
0answers
26 views

Existence of Linear Maps and the Fundamental Theorem of Linear Maps.

Prove that there does not exist a linear map $T: \Bbb R^5 \to \Bbb R^5$ such that $\operatorname{range}(T) = \operatorname{null} (T)$. My proof goes like this: Suppose for the sake of contradiction ...
0
votes
0answers
18 views

Nonlinear Maps with additivity or homogeneity

Examples of linear maps from $\phi :R^2 \to R$ that has homogeneity but is not linear. Example of a function $\phi : C \to C$ that is additive but is not linear. All the examples I have found for ...
0
votes
1answer
31 views

Finding and proving a basis for $W=\{f(x) \in P_2[\mathbb{R} ]:f'(x) +xf(0) = 0 \}$

I'm having a trouble proving/finding a basis for $W= \{f(x) \in P_2[\mathbb{R}]:f'(x) +x \bullet f(0) = 0 \}$. I'm supposing $\{ x, 1 \}$ is a basis for W because any vector in $P_2[\mathbb{R}]$ gets ...
0
votes
4answers
63 views

Find the basis for the subspace of the set of polynomials of degree less than five?

Let U = {p $\in P_4(F): p(2) = p(5) = p(6)$. Find a basis for U. I know how to do this problem if I were given p(2) = p(5). Set the two equal to each other and solve for one of the coefficients. I ...
0
votes
2answers
45 views

Matrix with eigen values given find [on hold]

Let$$ P=\begin{bmatrix} 0&-2&-3 \\ -1&1 &-1 \\ a&2 &b \end{bmatrix},$$ for some $a,b \in \mathbb{R}.$ Suppose that $1$ and $2$ are eigenvalues of $P$ and $$ ...
3
votes
0answers
53 views

Show that a certain operator is symmetric

I am trying to prove that the operator $L^2 = -\partial_\theta^2 - \cot\theta\,\partial_\theta - \frac{1}{\sin^2\theta}\partial_\phi^2$ fulfills the following property: For $y_{l,m} = ...
1
vote
0answers
35 views

having trouble with a 3-dimensional basis-change problem/

Let $V$ be a 3d vector space with a chosen basis $\alpha=\{e_1,e_2,e_3\}, \beta=\{f_1,f_2,f_3\}$ for $V$ satisfying: $$\begin{align}e_1 & =f_1+f_2+f_3 \\ e_2 &=f_2+2f_3 \\ e_3 & =f_3 ...
0
votes
2answers
37 views

i am having trouble with one of the homework question regarding to linear algebra(vector and span)

$V$ is a vector space of some dimension, with $\vec u,\vec v,\vec w$ independent set of vectors in $V$. define the subspace of $V$ given by $W = \operatorname{span}(\vec u-\vec v+\vec w, 2\vec u+\vec ...
0
votes
1answer
16 views

How to write a polynomial basis with conditions

I don't understand how to do problem where you have to write a basis for a polynomial. For a example a typical problem would be something like: Let U = {p $\in$ $P_n(F)$: p(2) = p(5) or p''(1) = ...
1
vote
1answer
47 views

Is this sufficient for linear independence proofs??

I've been doing all of these proofs the same basically, I just want to make sure I'm doing them right, I didn't include all the details but I have the outlines of my proofs here. 1) U and W are ...
0
votes
2answers
29 views

Least squares approximation: Legendre polynomial

Find the best quadratic least squares approximation to $f(x)=e^x$ on $[-1,1]$ with respect to the inner product $\langle f(x),g(x) \rangle = \displaystyle\int_{-1}^1 f(x)g(x)dx$. I cannot figure out ...
0
votes
2answers
49 views

Linear algebra proof

Let $W$ be a subspace of $\mathbb{R}^n$. Let $\vec{v}_1 ,\vec{v}_2 \in \mathbb{R}^n$. Suppose that $\vec{p}_1$ is the projection of $\vec{v}_1$ onto $W$ and $\vec{p}_2$ is the projection of ...
0
votes
1answer
36 views

Orthogonality question

Been stuck on this one: If $\vec{x}$ is orthogonal to $\vec{u}$ and $\vec{v}$ then $\vec{x}$ is orthogonal to $\vec{u}-\vec{v}$. Any hints?
0
votes
2answers
36 views

Can a set of 4 vectors with 3 entries each only span R2 if the third row reduces to all zeros?

I'm a bit confused as to how dimension, dimension of span, and dimension of column space all relate with regards to a basis. The question is worded as follows: Find the dimension of the span of the ...
1
vote
1answer
72 views

How to solve this graphing question?

$ \frac{|x-2|} {(x^2-4)}+\frac{(x-2)} {|x-2|} = b $ determine for which values of $b$ the equation has one and only solution. I tried sketching the graph, but was unable to do so accuratly...also, ...
0
votes
0answers
56 views

Showing that two sums are equivalent

given \begin{gather} U_d(x,y,q\mid i_1,\ldots,i_k)=\sum\limits_{n,m\geq0}x^ny^m\sum\limits_{\sigma = i_1\ldots i_k\sigma_{k+1}\ldots\sigma_m\in C_{[d]}(n,m)}q^{v(\sigma)}. \end{gather} show ...
1
vote
1answer
27 views

Direct Sum of Three Subspaces

Suppose $U = \{(x, y, x+y, x -y, 2x) \in \Bbb F^5 : x, y \in \Bbb F\}$. Find three subspaces $W_1, W_2, W_3$ of $\Bbb F^5$, none of which equal $\{0\}$ such that $\Bbb F^5 = U \oplus W_1 \oplus W_2 ...
0
votes
0answers
40 views

Copy of C in H , trace is independent of the choice

Let X€ Mn(H). For each of the choices of a copy of C in H , write out the corresponding matrix of X as an element of M(2n,C). Use this formula to show that the trace of X is independent of the choice. ...
2
votes
1answer
30 views

Given $A_{m\times n}$ and $B_{n \times m} (m<n)$. prove that AB is not singular and BA is singular

I have the following question which I can't seem to wrap my head around. I don't see how we can determine the desired just from the given info. Given $A_{m\times n}$ and $B_{n \times m}$ ...
0
votes
0answers
34 views

show by using leibniz formula

There are given $ r, s,n \in\mathbb N$ and $r+s=n$. It also given $A \in M_{r,K} $, $B \in M_{r\times s,K} $ and $C \in M_{s,K} $. Let $M$ be the matrix $\begin{bmatrix}A & B\\0 & ...
0
votes
1answer
23 views

Cramer's rule and linear dependence/independence test

When you have the system of equations: $$ax + by = e\\cx + dy = f$$ And you do some row operations to eliminate $y$, we get: $$x = \frac{ed-bf}{ad-bc}\tag{1}$$ If we eliminate $x$ we get: $$y = ...
1
vote
3answers
38 views

Determine whether or not a set is linearly independent

Prove or give a counter example: if $v_1, ..., v_n$ is linearly independent, is $5v_1-4v_2, v_2,...v_m$ also linearly independent. I'm not sure how to go about this. I tried a couple ways to prove ...
2
votes
2answers
52 views

Finding a basis for a set of polynomials

Let $U = \{p \in P_4(\mathbb{R}): p''(6) = 0\}$. Find a basis for $U$, then expand that basis to be a basis of $P_4(\mathbb{R})$. So I've been trying to find examples on how to approach this. I am ...
0
votes
3answers
67 views

Let $T : M_{\text{2x2}} (\mathbb{R}) \rightarrow \mathbb{P}_2$ be a linear transformation give by … [closed]

Hey guys could someone who is good at this take a look and tell me if I did it right =) I did all the work so it shouldn't take long to verify my results ... Thank you in advance Problem: Let ...
0
votes
3answers
24 views

In each part, find a basis for the given subspace ofR 3 , and state its dimension

guys I gotta be honest, I've taken notes on everything in the last two sections for this but I'm not sure how to find a basis for a subspace that is a lone plane/line etc.. a full explanation would ...
0
votes
1answer
55 views

Seeking Help on Linear Algebra Problem, Thank you all (YES or NO ANSWER)

Did i do the 1st one correct? Having difficulty understanding 2...1 step at a time just want to know if i did 1 correct
0
votes
1answer
20 views

the volume of pyramid value

when calculating the volume of pyramid using a determinnat, is it ok to take the determinanat in absloute value so that every negative result would be converted to positive volume number?
-2
votes
0answers
34 views

Finding the general solution for a nonhomogeneous linear equations.

Let $\quad {a}_{i},{b}_{i} \left(i=1,2,\dots,n \right)\in \mathbb{K},\mathbb{K}$ is a Field. $$\begin{cases} & \text ...
0
votes
1answer
27 views

Proving Linear Dependency of A based of $ (SpA)^\perp = \{(a,a-b,b-c,a)\mid a,b,c \in R\} $

I have this question for homework: Let A be a set of 2 vectors in $R^4$ Given that: $ (SpA)^\perp = \{ (a,a-b,b-c,a)\mid a,b,c \in R \} $ Prove that A is Linearly Dependent. I think I ...
0
votes
3answers
81 views

Complex Roots and calculations

roots of the equation $z^6 =1-\sqrt3 i $ are $$z_1,z_2,z_3,z_4,z_5,z_6 $$ calculate:$$|z_1|^3 +|z_2|^3+|z_3|^3+|z_4|^3+|z_5|^3+|z_6|^3$$ also calculate: $$z_1^6 +z_2^6+z_3^6+z_4^6+z_5^6+z_6^6$$ ...
0
votes
1answer
25 views

Calculation of eigenvalues in a Markov Chain

I'm trying to solve this exercise: Finding the eigenvalues of $A=(p_{ij})$ where $\sum_{i=1}^3 p_{ij}=1$ for all $j=1,2,3.$ In the $2 \times 2 $ case have $\lambda_1=p_{11}+p_{22}-1$ and ...
1
vote
0answers
25 views

$SO(n)$ is connected, alternative form

I have the following exercise: Show that $SO(n)$ is connected, using the following outline: For the case $n = 1$, there is nothing to show, since a $1\times 1$ matrix with determinant one must be ...
2
votes
1answer
44 views

Vector calculation question

the points a b c d are concordantly ( 1,2,-3) , (-1,2,1) , ( 0,1,-2) , ( 2,-1,1) find formula of the plane going thorugh d and which is pararlel to plane abc calculate the volume of pyramid abcd. ...
0
votes
2answers
21 views

Identify Orthogonal Proj. and Reflection within given choice of Matrices.

The problem states that out of five given matrices, one represents an Orthogonal Projection onto a line and another a Reflection about a line; I'm supposed to identify them. Rather than list the ...
4
votes
2answers
323 views

Eigenvalues and Eigenspace Question

Thank you ahead of time for the help, I am having a problem with part $4$. I understand parts $1$ and $2$ and $3$ and have solved them but I cant seem to understand $4$. If someone could help me out, ...
-3
votes
1answer
42 views

Linear Transormation T - please help so i can get some sleep =( [closed]

I am sorry for reasking these question but the other post, not sure what happened but it wouldnt let me reply anymore so i am reposting the question...i am really sorry its 7 am and i want to get some ...
1
vote
2answers
39 views

Does it span $\mathbb{R}^3$?

I have a T/F question and I think I know the direction to go, but I am not sure. It states: $\{[17,6,-4]^t,[2,3,3]^t,[19,9,-1]^t\}$ does not span $\mathbb{R}^3$. Let me get this straight. It SPANS ...
2
votes
1answer
58 views

Determining whether or not a vector is a linear combination of a give matrix

$$ A= \begin{bmatrix} 1 & 0 & 5\\ -2 & 1 & -6\\ 0 & 2 & 8 \end{bmatrix} ,b= \begin{bmatrix} 2\\ -1\\ 6 \end{bmatrix} $$ The problem asks to determine whether or not vector $b$ ...
1
vote
1answer
36 views

Are these two matrices equivalent?

I am supposed to row reduce a matrix to reduced row echelon form. $$ \begin{bmatrix} 1 & 2 & 4 & 8\\ 0 & 0 & 1 & 4\\ 0 & 0 & 0 & 0 \end{bmatrix} $$ I have tried ...
1
vote
1answer
63 views

Compute the determinant $4\times 4$

Compute the determinant: $$ A= \begin{vmatrix} 1 & 1 & a+1 & b+1 \\ 1 & 0 & a & b \\ 2 & b & a & b \\ 2 & a & a ...
0
votes
1answer
24 views

Span and linear independence of four vectors in $\mathbb{R}^3$

For every $u,v,t,w \in \mathbb{R}^3$, is it necessarily true that $\mathbb{R}^3 = \{u,v,t,w\}$ $t \in \operatorname{Sp}\{u,v,w\}$ $\{u,v,t,w\}$ is linearly dependent My answers: For 1: Not ...
6
votes
1answer
98 views

Existence of $p \times p $ matrices $A$ and $B$ over the field $\mathbb F_p$, $p$ prime, such that $AB-BA=I$. [duplicate]

Let $p$ be a prime number. Prove or disprove that there exists $p\times p$ matrices $A$ and $B$ over a field $\mathbb F_p$ with $AB-BA = I$. With the aid of MAPLE i was able to find out that ...
0
votes
1answer
35 views

Is there a subspace of $M^R_{2x3}$ that is isomorphic to $R_4[x]$?

Is there a subspace of $M^R_{2x3}$ that is isomorphic to $R_4[x]$? For example, Can I say that $M^R_{2x2}$ is a subspace of $M^R_{2x3}$ so it can be isomorphic to $R_4[x]$ ? (because they have ...
1
vote
2answers
48 views

Weird field notation

I have a question: Let $\mathbb{F}$ be any field characteristic $0$. Recall that $x_i$, denotes the $i^{th}$ entry of a vector $x\in\mathbb{F}^n$. Define $$S = \{x\in\mathbb{F}^5 \mid x_i = ...
1
vote
1answer
22 views

Solving $x' = Ax$ for real $x$ where $A$ is a matrix with complex eigen values

I have the following linear differential equation system: $$x' = A x$$ where $$ A = \left( \begin{array}{ccc} 1 & 0 & 0 \\ 3 & 1 & -2 \\ 2 & 2 & 1 \end{array} \right) $$ I ...
0
votes
1answer
28 views

Gauss-Jordan Method

I keep getting the wrong set of solutions can someone help me. I know that when using the Gauss-Jordan method, the rules that I must follow can be applied in a variety of different procedures then why ...
0
votes
2answers
54 views

Find conditions on a, b, c, and d for which the following system has solutions:

Find conditions on $a$, $b$, $c$, and $d$ for which the following system has solutions: $$2x+4y+z+3w=a $$ $$-3x+y+2z-2w=b $$ $$12x+5y-4z+12w=c $$ $$13x+10y-z+13w=d$$ I got the system down to: ...