1
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2answers
38 views

Invertible Proof with transposed matrices

Let A, B, C be square matrices that are invertible. Say I want to express X with no inverses Say $$ (A^{T}A)^{-1}(X +B^ {T})(C^{-1}B^{-1})^{T} = I. $$ I know that $A^{T}A$ = $I$, but where can I go ...
0
votes
1answer
25 views

Proof Writing Help: $P_UT=TP_U \Leftrightarrow U$ and $U^{\perp}$ are $T$-Invariant

I'm studying linear algebra using Axler's book on my own and this is also my first rigorous encounter with proofs would greatly appreciate suggestions to improve the writing of the first part of my ...
1
vote
1answer
32 views

Proof that P is an Orthogonal Projection

I'm studying linear algebra using Axler's text and am stuck on 6.17. The problem is: Prove that if $P \in \mathcal{L}(V)$ is such that $P^2 = P$ and every vector in $\operatorname{null} P$ is ...
2
votes
2answers
60 views

Prove $\dim W \ge 2$

Let $U_1, U_2, W$ subspaces of a finite dimensional vector space, such that: $U_1 \cap U_2 = \{0\}$ $U_1 \cap W \ne \{0\}$ $U_2 \cap W \ne \{0\}$ Show that $\dim W \ge 2$. ...
0
votes
0answers
22 views

Finding matrices values

I was trying to teach myself some things about saddle points. This is a little more advance when it comes to finding the number for each matrices and a value $v$ $($say $v=(1/3), (1/3), (1/3))$ ...
0
votes
0answers
29 views

Matrix Saddle Points and Dominance

I was teaching myself about dominance relations and saddle points after a friend of mine started discussing it with me and how it can be used in games. I wanted to know how to prove a problem like ...
2
votes
4answers
62 views

Prove that if product of matrices is singular, one of the matrices is singular.

I'm having trouble with this proof, it would be much easier to work out the other way it seems. Let $A$ and $B$ be square matrices of equal size. Prove that if $\det(AB) = 0 =C$ then either $A$ or ...
2
votes
1answer
44 views

Question about $e^T$ where T is a transformation

First off, I'm given a matrix $A$ s.t. the characteristic polynomial of $A$ is $p(a) = (-1)^nx^n+x^2-x+2$ and am asked to find $det(A^k)$ for a natural $k$ and $det(e^A)$. So from the polynomial I get ...
2
votes
2answers
21 views

Mustn't both left and right inverses be checked? [Lay P160 Ch 2 Sup Q4]

Question: Suppose $A^n = 0$ matrix for some $n > 1$. Find an inverse for $I - A$. Solution: From P160 Supplementary Exercise 3, the inverse of $I-A$ is probably $I+A+A^{2}+...+A^{n-1}$. To ...
0
votes
4answers
64 views

Integral Problem $\sin^6 x$. [closed]

What is the integral of $\sin^6 x$? Can some one publish it with the method it would be really helpful. Specially I want the trig identities.
10
votes
4answers
174 views

Proof: Show there is set of $n+1$ points in $\mathbb{R}^n$ such that distance between any two distinct points is $1$?

Argh, I hate to ask a question again so soon, especially one I feel like I should know. Linear algebra is taking its toll, and I am not quite used to the theory side of mathematics. Anyways, I ...
0
votes
2answers
50 views

Proof that if we add a vector to a linearly dependent set of vectors in a vector space $V$, then the new set of vectors is still linearly dependent

Prove that if $S=\{v_1, v_2, v_3\}$ is a linearly dependent set of vectors in a vector space $V$, and $v_4$ is any vector in $V$ that is not in $S$, then $\{v_1, v_2, v_3, v_4\}$ is also linearly ...
1
vote
1answer
43 views

Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W

Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W This is something from a practice sheet I got. I'm studying for a linear algebra final. I am unsure if we have ...
2
votes
4answers
149 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 ...
1
vote
3answers
89 views

Prove determinant of $n \times n$ matrix is $(a+(n-1)b)(a-b)^{n-1}$? [duplicate]

Prove $\det(A)$ is $(a+(n-1)b)(a-b)^{n-1}$ where $A$ is $n \times n$ matrix with $a$'s on diagonal and all other elements $b$, off diagonal.
0
votes
2answers
48 views

Sum of the eigenvalues

if $V$ is a finite-dimensional vector space and $t \in \mathcal L (V,V) $is such that $t^2 = id_V$ prove that the sum of eigenvalues of t is an integer. I started the prove as such: Let $\lambda_1 ...
2
votes
2answers
63 views

Could this linear algebra proof be done without computation?

From page 95 of Hoffman & Kunze's Linear algebra: Let $T$ be the linear operator on $\mathbb{R}^2$ defined by $T(x_1,x_2)=(-x_2,x_1)$ Prove that if $B$ is any ordered basis ...
3
votes
0answers
69 views

Proof for the form of characteristic polynomial

I'd like to proof: The caracteristic polynomial of $A \in M(n\times n, K)$ has the form: $P_A(\lambda) = (-1)^n \lambda^n + (-1)^{n-1} \operatorname{tr}(A)\lambda^{n-1} +\dots +\det(A)$ My proof ...
0
votes
1answer
75 views

Proof: $ker(g) \subset ker(f)$ ..

Let $V = \mathbb{R}_{\le 3} [x]$ with basis $ B = (1, x, x^2, x^3)$. And $f: V \to \mathbb{R}, p \to \int_{-1}^1 p(x) dx$ and $g: V \to \mathbb{R}^3, p \to ^t( p(-1), p(0), p(1) )$. (1) I had to ...
1
vote
0answers
20 views

Show equivalence corresponding Nulls of function.

I'd like to show that the following two propositions are equivalent: (1) $f \in \mathbb{R}[x]$ has a multiple Null, so it's $\ge 2$. (2) $f$ and $f'$ have a common Null, whereas $f'$ describes the ...
1
vote
1answer
40 views

show that for any $n \in \mathbb{Z}$ gcd($n^2 - n + 1, n +1)$ is either $1$ or $3$.

show that for any $n \in \mathbb{Z}$ gcd($n^2 - n + 1, n +1)$ is either $1$ or $3$. My Work: I considered the case where $n =-1$ , and the case $n \not= 1$. So when $n\not= -1$ we can let $n^2 - ...
1
vote
0answers
73 views

Proving $x_i = \dfrac{\det(A_i)}{\det(A)}$

Suppose we have an nxn matrix A, and $A\underline{x} = \underline{p}$ prove that $x_i = \dfrac{\det(A_i)}{\det(A)}$ where $A_i$ is the matrix obtained from A by replacing the $i^{th}$ column by ...
0
votes
1answer
67 views

Differential map on the vector space of polynomials: Kernel and Image

Given the $V_n$ is the vector space of polynomials of degree $\leq$ n over $\Bbb R$ So $M_D = \begin{bmatrix} 0 & 1 & ... & 0\\ 0 & 0 & 2 &... 0 \\ 0 & 0 &0 &.\\ . ...
0
votes
0answers
28 views

Understanding 2nd half rank-nullity theorem proof.

I'm trying to understand the second half of the rank-nullity theorem (the part that shows $T(e_{k+1}) \dots T(e_{k+r})$ is independent). Assume $e_1 ,\dots e_k, e_{k+1}, \dots e_{k+r}$,is a basis for ...
0
votes
0answers
33 views

Linear algebra proof help

Let $A$ and $B$ be similar matrices. Prove that the geometric multiplicities of the eigenvalues of $A$ and $B$ are the same. [Hint: show that, if $B=P^{-1}AP$, then every eigenvector of $B$ is of the ...
5
votes
0answers
220 views

Span and Dimension: A subspace

If $A$ is finite set of linearly independent vectors then the dimension of the subspace spanned by $A$ is equal to the number of vectors in $A$. This is obviously true. Since $A$ is a finite set of ...
1
vote
1answer
60 views

Linear Independent proof

In my Linear Algebra class we define Linear dependence as follows: If $F$ is a field and $V$ is a vector space over the field $F$. The set $A = {\lbrace v_1,v_2,...,v_k \rbrace}$ where ...
1
vote
1answer
34 views

Proving Direct Sum

Claim. Let $V$ be a vector space over $F$, and suppose that $W_1$, $W_2$, and $W_3$ are subspaces of $V$ such that $W_1 + W_3 = W_2 + W_3$. Then $W_1 = W_2$. I know that this claim is false, but ...
1
vote
5answers
118 views

Showing something is not onto?

Quick question..: If I have a linear transformation $T:\mathbb R^2 \to \mathbb R^4$, can I say that since $\mathbb R^4$ is greater than $\mathbb R^2$, it can never be onto - since the elements in ...
0
votes
1answer
51 views

proof $B^{-1}MB$ is triangular

How to prove this? Theorem: Let $M$ be a matrix of complex numbers. There exists a non-singular matrix $B$ such that $B^{-1}MB$ is a triangular matrix. This is corollary from book Linear Algebra by ...
1
vote
1answer
23 views

Linear Algebra Proof with one-dimensional subspaces

Suppose that V is finite dimensional, with $dimV=n$. Prove that there exist one-dimensional subspaces $U_1,...,U_n$ of $V$ such that $$V = U_1 \oplus\dotsb\oplus U_n$$ My linear algebra is rusty, very ...
1
vote
4answers
91 views

Prove or Disprove the existence of a basis

I'm asked to prove or disprove the existence of a basis $(p_0,p_1,p_2,p_3)$ of $F(t)(3)$ (Polynomials of degree at most 3) such that each of the polynomials $p_0,p_1,p_2,p_3$ satisfies the equation ...
0
votes
1answer
25 views

Fixpoints of affine transformations

I want to find out all the possibilities what fixpoints of an affine transformation can be in 2-dim vector space. If the transformation is identity, then it is trivial - fixpoints describe the ...
0
votes
1answer
146 views

Proof that the set of all invertible $n \times n$ matrices of real numbers is a vector space.

I'm studying Algebra and I'm asked to prove or disprove the statement above. I found that is true, but I'm not sure how to prove it. My problem here is that this statement is too "broad", i.e. I ...
1
vote
1answer
57 views

Given $A$ and $\vec{b}$ in $A\vec{x}=\vec{b}$, solve for $\vec{x}$

What are the steps to solve for $\vec{x}$, given that $A\vec{x}=\vec{b}$ and we know what $A$ and $\vec{b}$ are? I know the first thing you do is multiply each side by $A^T$ ...
12
votes
4answers
331 views

Tips for writing proofs

When writing formal proofs in abstract and linear algebra, what kind of jargon is useful for conveying solutions effectively to others? In general, how should one go about structuring a formal proof ...
0
votes
1answer
59 views

Linear algebra hw! Linear transformation

Let $T : V -> V$ be a linear transformation where V is a nite dimensional vector space. If rank(T) = rank$(T^2)$, prove that image(T)$\cap$Ker(T) = {0}. I have to give this hw to my prof this ...
3
votes
1answer
102 views

Proving there are infinitely many integers having the identical set of prime factors.

Let positive integers $a$ and $b$, and let $a_0, a_1, a_2 \ldots$ where $a_i = a + b*i$ is the infinite arithmetic sequence they determine. Prove that there are infinitely many $a_i$ having the ...
1
vote
1answer
44 views

Linear algebra: Matrix multiplication problem

I need to prove something in my homework I just don't know how to approach it and need some guidance. "Show that for a matrix $A$ ($n \times m$) and a vector $\vec{x}$ ($m \times 1$) it applies that: ...
0
votes
2answers
48 views

Prove that $R(P) = N (I − P) = X$ and $R(I − P) = N (P) = Y$.

Suppose that $V = X ⊕Y$, and let $P$ be the projector onto $X$ along $Y$. Prove that $R(P) = N (I − P) = X$ and $R(I − P) = N (P) = Y$. I know that from $V = X ⊕Y$ I got $v=x+y$ for $v,x,y$ are ...
0
votes
3answers
53 views

Where V is a Vector Space, $\forall \overrightarrow{v} \in V, 0\overrightarrow{v} = \overrightarrow{0}$

I'm denoting all vectors as such: $\overrightarrow{v}$. Any variable without an arrow above is a scalar. Suppose $V$ is a vector space over $F$, with additive identities $\overrightarrow{0}$ and $0$ ...
3
votes
2answers
538 views

Learning Proofs (for Computer Science)

Harvard's math curriculum, for freshmen, is divided into 4 classes beyond the BC Calculus level, Math 21, 23, 25 and 55. Math 21 is your classic plug-and-chug multivariable calculus and linear algebra ...
1
vote
0answers
83 views

Proving direct sums.

Let $V$ be a vector space over $F$ and let $U$ and $W$ be subspaces such that $V=U+W.$ Prove that $V=U\oplus W$ if and only if $U\cap W=\lbrace 0 \rbrace$. Attemp: Suppose $V=U\oplus W$. By ...
0
votes
2answers
86 views

Uniqueness Proof: Related to Division Algorithms

Regarding the statement: Let $a\in \mathbb {Z}$, $b\in \mathbb {Z}$. Then there exists integers q and r such that $a = qb+r$ where $0\le r \le b$. Let $S$ $=$ $\lbrace a-qb: q\in \mathbb {Z}, a-qb ...
1
vote
1answer
39 views

Why is it sufficient for a normal to only be orthogonal to 2 vectors on a plane instead of 3?

The following is an excerpt from my textbook: It is clear geometrically that there is a unique plane containing any 3 points $A,B$ and $C$ that are not all on a line. In determining the equation ...
0
votes
2answers
69 views

Prove the identity

$$\cos \frac{x}{2} \cdot \cos \frac{x}{4} \cdot \cos \frac{x}{8} = \frac{\sin x}{ 8\sin \frac{x}{8}}$$ Conjecture a generalization of this result and prove its correctness by induction. Ps: I have ...
4
votes
1answer
124 views

Endomorphism- Nilpotent matrices

An endomorphism $f: V \rightarrow V$ of an $F$-vector space is called nilpotent iff there exists $ \delta \in \mathbb N$ such that $f^\delta=0$. Suppose that $f : V\rightarrow V$ is a nilpotent ...
0
votes
1answer
70 views

How can we prove that an affine function can be written as $f(x)=ax+b$?

Sorry if duplicate. Searched but could not find anything. I understand the concept of an affine function. I am not sure as to how to go about writing a proof for the definition $f(x)=ax+b$ where ...
0
votes
1answer
41 views

I need help on manipulating this expression:

Assume that $(4k + 3) ^ 2 - (4k + 3)$ is not divisible by 4. If this is true, prove that $(4(k+1) + 3) ^ 2 - (4(k+1) + 3)$ is not divisible by 4. I need to prove this for my induction problem, and ...
1
vote
1answer
40 views

Properties for internal stability of a discrete-time system

These are two parts of a larger proof I'm working on, can't figure how i) implies ii) though. Dynamic system: $x_{(k+1)} = Ax_{k}, x(0)=x_0$ Where $A \in \mathbb{R}^{n\times n} $ is a real ...