1
vote
1answer
109 views

Showing that planes intersect

let there be two planes $$2x-y-5z+11=0$$ and$$2x+2y+z-1=0 $$ show that they intersect attempt at a solution: If planes do not intersect they are parralel hence there is a $t\in R$ such that ...
0
votes
0answers
31 views

What is the linear combination of B?

I have a problem where I am finding $A^n$B where B=$[3,1,1]^t$. I know the steps in solving, but I do not remember how to find linear combination. I do not see it. There has to be a way to calculate ...
1
vote
1answer
25 views

Prove existence of Diagonalizable Matrix

Suppose R, T $\in L(F^3)$ each have 2, 6, 7 as eigenvalues. Prove that there exists an invertible operator S $\in L(F^3)$ such that $R=S^{-1}TS$. What I got so far is that since R and T have three ...
0
votes
1answer
27 views

Diagonalization of Skew symmetric matrix

I have a skew symmetric matrix $$C=\left( \begin{array}{ccc} 0 & -a_3 & a_2 \\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \\ \end{array} \right).$$. and we have the relation $C=UDU^{-1} ...
0
votes
4answers
59 views

Give a Counterexample if V is infinite dimensional

$V = nullT \oplus rangeT$ if and only if $V = null T + rangeT$. Where $T \in L(V)$ I'm having alot of trouble coming up with an example for this. Shouldn't both cases always fail if V is infinite ...
0
votes
4answers
34 views

Prove that T = I with Linear Transformations.

Suppose that $T \in L(V)$ and $T^2 = I$ and -1 is not an eigenvalue of T. Prove that T = I. What I tried was: Suppose $\lambda$ is an eigenvalue of T such that $T(v) = \lambda v$ Then we know that ...
1
vote
2answers
26 views

Applying a polynomial to an operator?

Suppose $T \in L(V)$ and $\exists$ a positive integer n such that $T^n = 0$. Prove that $(I-T)$ is invertible and that $(I-T)^{-1} = I + T + \dots + T^{n-1}$. I wish I could say that I attempted ...
1
vote
2answers
27 views

Prove that these have the same eigenvalues

Suppose $T \in L(V)$. Suppose $S \in L(V)$ is invertible. Prove that $T$ and $S^{-1}TS$ have the same eigenvalues. What is the relationship between the eigenvectors of T and the eigenvectos of ...
0
votes
1answer
30 views

Why do these vectors not span the given space?

I need some help understanding this solution to a problem. I am working on the problem above. I know that in order for a set of vectors to be a basis it must be linearly independent and span the ...
0
votes
0answers
35 views

proof that T^k is a positive operator

so the book (Axler linear algebra done right) asks me to prove that if $T$ is a positive operator then $T^k$ is also positive , now the book defines a positive operator as an operator which is self ...
1
vote
1answer
22 views

Matrix of a linear map Questions

Suppose n < m. Show there exists a basis $w_1...w_m$ of w for every choice of basis for v of degree n such that the last m-n rows of M(T) consist of only $0$'s for every choice of basis for w. ...
0
votes
1answer
62 views

solving the equation

let there be a function $ f(x)= \ln x-kx^2, k>0$ determine for whihc values of $ k$ ,the equation $f(x)=0.5$ has a single solution; attemp to solve: $$0.5 = \ln x-kx^2$$ $$kx^2 +0.5 = \ln x $$ ...
0
votes
1answer
20 views

Matrix Rank calculation

I have a matrix A . A can be written as A=B+D. I know rank of B. It is 3. Is it possible for A to have ranks $<3$ . If so please prove.
-1
votes
0answers
18 views

Accuracy of line intersecting algorithem decrase with large precisions

from the above pic I found the value of x from equation of line p1-p2 and perpendicular line from point a to the Line(p1,p2) .The intersecting point is X ,but the accuracy is less see the result ...
-1
votes
2answers
28 views

Invertible Linear Maps Proof [on hold]

1) Suppose $V$ is finite dimensional and $S$, $T$, $U \in L(V)$ and $STU = I$. Show $T$ is invertible and $T^{-1} = US$. 2) Suppose $V$ is finite dimensional and $R$, $S$, $T \in L(V)$ are such that ...
0
votes
1answer
13 views

Linear Operators Injectivity and Surjectivity

Suppose T $\in L(P(R))$ is such that T is injective and deg Tp $\leq$ deg p for every nonzero polynomial p $\in P(R)$. Prove that T is surjective and that deg Tp = deg p for every nonzero p $\in ...
0
votes
1answer
25 views

shortest point on a line segment from point out side the line

from the above pic I found the value x from line (p1,p2) and point a using y=mx+b and imaginary red line which is perpendicular to black line having slope -1/m and the intersecting point x. the ...
2
votes
2answers
53 views

Decompose a real symmetric matrix

Prove that, without using induction, A real symmetric matrix $A$ can be decomposed as $A = Q^T \Lambda Q$, where $Q$ is an orthogonal matrix and $\Lambda$ is a diagonal matrix with eigenvalues of $A$ ...
0
votes
1answer
24 views

Linear Maps from a finite space to an infinite space

Suppose V is finite dimensional with dim V > 0. Prove that if W is infinite dimensional then $L(V, W)$ is infinite dimensional. Help? I really have no idea how to go about this one? I'm assuming I ...
0
votes
0answers
34 views

Linear maps and linearly independent sets

Suppose $v_1,...,v_m$ is a linearly dependent list of vectors in $V$. Suppose also $W \neq \{0\}$. Prove there exist $w_1,...,w_m$ in $W$ such that no $T\in L(V, W)$ satisfies $Tv_j = w_j$ for $j = ...
0
votes
2answers
18 views

Quick question on the basis of subset of polynomals

Let U = {p $\in$ $P_4(F)$: $p(2) = p(5) = p(6)$} Find a basis of U. So the way I did this problem was by writing out $p(2) = p(5)$ and $p(5) = p(6)$, then I made a system of equations and solved for ...
2
votes
2answers
35 views

Dimension of the sum of three subspaces

So I was doing practice problems in my textbook and I'm really stuck on this one: We know that $$\dim(U_1 + U_2) = \dim U_1 + \dim U_2 - \dim(U_1 \cap U_2)$$ if $U_1$ and $U_2$ are finite dimensional ...
2
votes
4answers
70 views

Proof involving linear maps

Suppose $V$ is a vector space and $S,\ T \in L(V)$ such that range $S \subset$ null $T$. Prove that $$(ST)^2= 0$$ I have no idea how to go about this could someone maybe explain it in English or ...
0
votes
1answer
24 views

Rank Nullity Theorem application

Show that {T $\in$ $L(R^5, R^4)$: dim null T > 2} is not a subspace of $L(R^5, R^4)$ I have no idea how to show this isn't a subspace the farthest I have gotten is to show that dim range T < 3 ...
1
vote
0answers
18 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
27 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
33 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
22 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
32 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
65 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 ...
-1
votes
2answers
47 views

Matrix with eigen values given find [closed]

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
57 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
30 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
50 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
57 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
42 views

Copy of C in H , trace is independent of the choice [on hold]

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
32 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
53 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 ...